Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space

HYPERSPHERE-BASED RESTRICTING CONDITIONS F OR COLORINGS OF THE EUCLIDEAN SP A CE GABRIEL ISTRA TE AND CA T ALIN ZARA Abstract. W e in v estigate colorings of the Euclidean space R n in whic h the color of a p oin t p is determined by mono c hromatic configurations of p oin ts lying on h yp erspheres cen tered at p . W e consider t wo t yp es of conditions: those based on the cardinalit y of the forcing sets and admis- sible radii, and those based on certain geometric prop erties of simplices, suc h as shape, edge lengths, or v olumes, for colorings using coun tably (fi- nite or infinite) man y colors. Our main ob jective is to determine whether a giv en condition forces the coloring to be monochromatic or not. Exam- ples constructed using existing results sho w that, for conditions based on shap es, additional regularity assumptions on color classes are necessary . Accordingly , w e study colorings that are somewhere comeager. Contents 1. In tro duction 1 2. Cardinalit y Conditions 4 3. Rigid Geometric Prop erties 6 4. Somewhere Comeager Colorings 8 5. Simplex Shap e Conditions 12 6. Edge-Length Conditions 13 7. V olume Conditions 15 References 16 1. Introduction In this pap er w e in v estigate colorings of the n − dimensional Euclidean space R n , with n ⩾ 2, for whic h the color of a p oin t p is constrained by mono c hromatic sets of p oin ts on hyperspheres centered at p . A c oloring of a top ological space X is a function f : X → C from X to a nonempt y set C . F or eac h c ∈ C , the set f − 1 ( c ) of p oin ts of c olor c is called the c olor class of c . A coloring is mono chr omatic if f is constan t on X . It is lo c al ly mono chr omatic if every point has an op en neigh borho o d on which f is constan t, and it is somewher e lo c al ly mono chr omatic if f is constan t 2020 Mathematics Subje ct Classific ation. Primary 52C10. Secondary 05D10, 54E52. Key wor ds and phr ases. geometric Ramsey theory , space colorings, forcing configura- tions, rigid motions, simplices, Baire category . 1 2 GABRIEL ISTRA TE AND CA T ALIN ZARA on some nonempty op en subset of X . The top ological spaces X considered in this paper are the space R n with the standard top ology induced b y the Euclidean distance, together with its top ological subspaces. The general framework is as follows. Let n ⩾ 2 b e a natural num ber, C ⊆ N , Ω ⊆ R n , and R ⊆ (0 , ∞ ) b e nonempt y sets. The elements of C are referred to as c olors , the elements of Ω as admissible c enters , and the elemen ts of R as admissible r adii . W e denote b y S r ( p ) the (hyper)sphere 1 with cen ter p and radius r , and call it an admissible spher e if p ∈ Ω and r ∈ R . Let P b e a fixed property that s ubsets of the space may or may not satisfy . W e study colorings f : R n → C that satisfy conditions of the follo wing t yp e. Q C , P , Ω , R : F or an y color c ∈ C and any admissible sphere S r ( p ), if the set f − 1 ( c ) ∩ S r ( p ) has a subset with prop ert y P , then f ( p ) = c . This condition sa ys that if an admissible sphere has a mono c hromatic subset satisfying the giv en prop ert y , then the center of the sphere m ust hav e the same color as that subset. The main question we address is whether a coloring satisfying a condition of type Q C , P , Ω , R m ust b e lo cally or globally mono c hromatic on Ω. This study was inspired by the follo wing problem, sligh tly paraphrased, whic h appeared as Problem B1 on the 86th W. L. Putnam Mathematical Comp etition in Decem b er 2025, see [1]. Supp ose that e ach p oint in the plane is c olor e d either 1 (r e d) or 2 (gr e en), subje ct to the fol lowing c ondition. If a cir cle c ontains thr e e p oints of the same c olor c ∈ { 1 , 2 } , then the c enter of the cir cle also has c olor c . Show that al l p oints in the plane have the same c olor. In the framework describ ed ab o ve, we hav e n = 2 (t w o-dimensional space), C = { 1 , 2 } (tw o colors), Ω = R 2 (all p oin ts are admissible cen- ters), and R = (0 , ∞ ) (all radii are admissible). The prop ert y P is that “ the set has c ar dinality at le ast 3.” One of the solutions presen ted in [1] considers a slightly more general setting. It allows an arbitrary finite num b er X of colors and requires that the intersection con tain at least a fixed n um ber Y of p oin ts in order to force the conclusion ab out the color of the cen ter. In this case C = { 1 , . . . , X } , and the prop ert y P is that “ the set has c ar dinality at le ast Y .” W e denote the corresp onding condition Q C , P , Ω , R b y Q X,Y , Ω , R . The conclusion remains the same. If all circles are admissible, that is if Ω = R 2 and R = (0 , ∞ ), then any coloring f satisfying Q X,Y , Ω , R is mono c hromatic. 1 F or conv enience, unless sp ecified otherwise, w e will use the term spher e to mean hyp erspher e . HYPERSPHERE-BASED RESTRICTING CONDITIONS 3 In Section 2 we extend the colorings from the plane to higher dimensional spaces R n , the cardinalities X and Y from finite to countable 2 , restrict R to an arbitrary uncoun table set, and allow Ω to be any nonempt y subset of R n . The main result of the section, Theorem 2.1, states that an y coloring f satisfying Q X,Y , Ω , R is lo cally mono c hromatic on Ω, hence, if Ω is a connected top ological space, mono c hromatic on Ω. In subsequent sections, w e study geometric prop erties P that are inv arian t under rigid motions of the space. In Section 3 we discuss general prop erties of the group of rigid motions, define rigid ge ometric pr op erties , and intr o duce sev eral classes of examples, using geometric prop erties of simplices in R n . Beginning in Section 4 we imp ose a regularity condition and consider somewher e c ome ager c olorings f : R n → C . These are colorings with the prop ert y that there exists a color class that is comeager in some op en subset. W e assume familiarity with basic concepts and results from Baire category theory , for example as presen ted in [10]. In particular, w e rely on the defini- tions and basic prop erties of meager, nonmeager, comeager, and Baire sets in second-countable topological spaces. W e allow all points to be admissible cen ters and restrict the set of ad- missible radii to sets that are comeager near 0. The main general result of Section 4 is Theorem 4.6, whic h giv es sufficient conditions on prop er- ties P under whic h the colorings under consideration are somewhere lo cally mono c hromatic or globally mono c hromatic. F rom this theorem we deduce the main general result for colorings satisfying rigid geometric prop erties, Theorem 4.7. In the remaining sections w e apply Theorem 4.7 to properties P defined by classes of simplices. Those conditions are given in terms of shap es (Section 5), edge lengths (Section 6), or volumes (Section 7). In Section 5 we consider prop erties based on simplex shap es, namely isosceles, regular (equilateral), and right simplices. In Theorem 5.1 w e sho w that in all three cases the colorings under consideration are somewhere mono c hromatic. W e construct an example demonstrating that colorings sat- isfying the conditions based on regular (for all n ⩾ 2) or righ t simplices (for n = 2) need not b e mono c hromatic. F or this example the color classes are con v ex subsets and the construction uses the fact that, for those simplices, the circumcenter lies in the conv ex hull of the vertices. By contrast, no suc h limitation arises for isosceles (for all n ⩾ 2) or right (for n ⩾ 3) sim- plices, and in those cases we prov e that the colorings under consideration are mono c hromatic. In Section 6 w e consider a differen t class of properties P , defined in terms of a set L of admissible edge lengths. If w e require only that at least a certain n um b er of edge lengths of a simplex b elong to L , but not necessarily all edge lengths, then greater flexibility is a v ailable. In this setting, Theorem 6.1 sho ws that all colorings under consideration are mono c hromatic if and only if inf ( L ) = 0. By con trast, requiring al l edge lengths to lie in L imp oses 2 w e use the term c ountable to mean the cardinality of a subset, finite or infinite, of N 4 GABRIEL ISTRA TE AND CA T ALIN ZARA greater rigidity , and the infimum condition alone is no longer sufficien t, as sho wn in Example 6.2. In Theorem 6.3 we sho w that, in this case, a sufficient condition is the stronger requiremen t that L b e comeager near 0. In Section 7 w e consider conditions based on volumes of simplices and sets V of admissible v alues. In Theorem 7.1 w e prov e that all colorings under consideration are mono c hromatic if and only if inf ( V ) = 0. 2. Cardinality Conditions In this section w e show that the conclusion of the Putnam comp etition problem remains v alid in more general settings. These include v ariations in the finite dimension of the space, the cardinalit y of the set of colors, the minim um num ber of same-color points on a circle required to force the color of the cen ter, the cardinality of the set of admissible radi i, and the c hoice of admissible centers. In the general framework, let Y ⩽ ℵ 0 b e a fixed cardinalit y . W e consider the following property . P ( Y ) : A set has this prop ert y if its cardinality is at least Y . W e refer to this prop ert y simply as ( Y ). Theorem 2.1. L et R ⊂ (0 , ∞ ) b e an unc ountable set of admissible r adii, let C b e a set of c olors of c ar dinality X ⩽ ℵ 0 , let Y ⩽ ℵ 0 b e a fixe d c ar dinality, and let Ω ⊂ R n b e a nonempty set of admissible c enters. L et f : R n → C b e a c oloring of the sp ac e R n satisfying the fol lowing c ondition: Q X,Y , Ω , R : F or every c olor c ∈ C and every admissible spher e S r ( p ) , if the set f − 1 ( c ) ∩ S r ( p ) has c ar dinality at le ast Y , then f ( p ) = c . L et f | Ω : Ω → C denote the r estriction of f to the induc e d top olo gic al sp ac e Ω . Then f | Ω is lo c al ly mono chr omatic. If Ω is c onne cte d, then f | Ω is mono chr omatic. Pr o of. Since the uncoun table set R = [ m ∈ Z  [2 m − 1 , 2 m ] ∩ R  is a coun table union, at least one of these subsets must b e uncountable. Hence, there exists a = 2 m − 1 > 0 suc h that [ a, 2 a ] ∩ R is uncountable. Let p ∈ Ω b e an admissible center and let B = B 2 a ( p ). If B ∩ Ω = { p } , then { p } is op en in Ω, and f is lo cally mono c hromatic at p . Otherwise, assume that B ∩ Ω  = { p } . Cho ose q ∈ B ∩ Ω with q  = p , and let d = d ( p, q ) ∈ (0 , 2 a ) denote the distance b et ween p and q . Cho ose m ∈ N HYPERSPHERE-BASED RESTRICTING CONDITIONS 5 suc h that d > a/m = ϵ . Since the uncoun table set [ a, 2 a ] ∩ R = m [ k =1  [ a + ( k − 1) ϵ, a + k ϵ ] ∩ R  is a finite union, at le ast one of the subsets m ust b e uncoun table. Therefore, there exists b ∈ [ a, 2 a ) suc h that R ∩ ( b, b + ϵ ) is uncountable. Supp ose that n = 2, hence, that we are coloring the plane R 2 . Let I ⊂ R ∩ ( b, b + ϵ ) b e a countable subset. F or each r ∈ I , the circle C r ( p ) = S r ( p ) is admissible. F or any color c  = f ( p ), the set f − 1 ( c ) ∩ C r ( p ) m ust ha v e cardinality strictly less than Y , otherwise f ( p ) = c . Hence, the set A = [ r ∈ I [ c ∈C \{ f ( p ) }  f − 1 ( c ) ∩ C r ( p )  has cardinality at most X · Y ⩽ ℵ 0 , and is therefore coun table. Let D = { d ( q, T ) | T ∈ A } b e the set of distances from q to p oin ts of A . Then D is also countable. Since R ∩ ( b, b + ϵ ) is uncoun table, there exists an admissible radius R ∈ ( R ∩ ( b, b + ϵ )) \ D . Consider the circle C R ( q ). F or every r ∈ I , we ha v e | R − r | < ϵ < d ( p, q ) < 2 a ⩽ 2 b < R + r, so the circles C R ( q ) and C r ( p ) in tersect in exactly tw o points. Because C R ( q ) ∩ A = ∅ , all suc h intersection p oin ts hav e color f ( p ). Since I is coun table, it follows that   f − 1 ( f ( p )) ∩ C R ( q )   ⩾ 2 ℵ 0 = ℵ 0 ⩾ Y , whic h forces f ( q ) = f ( p ). Supp ose no w that n ⩾ 3. Let r, R ∈ R ∩ ( b, b + ϵ ). The admissible spheres S r ( p ) and S R ( q ) intersect in an ( n − 2) − sphere S , and S has uncoun table man y p oin ts. As a subset of S r ( p ), the sphere S can hav e at most countably man y p oin ts of color other than f ( p ), hence, S has uncoun table man y points of color f ( p ). Since S is also a subset of S R ( q ), it follo ws that f ( q ) = f ( p ). Th us, for ev ery p ∈ Ω, there exists an op en ball B ⊂ R n suc h that f ( q ) = f ( p ) for all q ∈ B ∩ Ω. Since B ∩ Ω is op en in Ω, the restriction f | Ω is lo cally mono c hromatic. If Ω is connected, then every lo cally constan t function on Ω is constan t. □ The next example sho ws that if Ω is not connected, then f | Ω ma y b e lo cally monochromatic without b eing globally mono c hromatic. 6 GABRIEL ISTRA TE AND CA T ALIN ZARA Example 2.2. Let C = { 1 , 2 } , let Y ⩽ ℵ 0 , let R = (0 , 1), whic h is uncoun t- able, and let Ω = B 1 (( − 2 , 0)) ∪ B 1 ((2 , 0)) . Define f to b e constantly 1 on B 2 (( − 2 , 0)), constantly 2 on B 2 ((2 , 0)), and arbitrary elsewhere in the plane. Then f satisfies Q X,Y , Ω , R , but takes dif- feren t v alues on the t w o connected comp onen ts of Ω. Hence, f | Ω is not mono c hromatic. □ The follo wing example sho ws that even when Ω is connected and f | Ω is mono c hromatic, the coloring f : R 2 → C need not b e somewhere locally mono c hromatic. Example 2.3. Let Q 2 denote the set of rational p oin ts in the plane R 2 , and let Ω = R 2 \ Q 2 . Define f : R 2 → { 0 , 1 } by f ( x ) = ( 1 , x ∈ Q 2 , 0 , x ∈ Ω . F or any X ⩾ 2, Y ⩾ 3, and R ⊂ (0 , ∞ ), the coloring f satisfies Q X,Y , Ω , R . Indeed, if a circle con tains at least three rational p oin ts, then its cen ter is also rational. 3 Consequen tly , if p ∈ Ω, then for ev ery r ∈ R , f − 1 (1) ∩ C r ( p ) has at most tw o p oin ts and f − 1 (0) ∩ C r ( p ) is infinite, so f ( p ) = 0. The induced topological space Ω is connected, in fact path-connected, and f | Ω is constan t, consistent with Theorem 2.1. Ho w ever, both color classes f − 1 (1) = Q 2 and f − 1 (0) = Ω are dense in R 2 , so f is not constant on an y nonempt y op en disk in R 2 . Note that if p ∈ Q 2 and r > 0, then f ( p ) = 1 and f − 1 (0) ∩ C r ( p ) is infinite. This do es not con tradict Q X,Y , Ω , R , since the condition applies only to circles with centers in Ω = R 2 \ Q 2 . □ 3. Rigid Geometric Proper ties In later sections w e will study properties P based on geometric conditions that are inv arian t under isometries, and in this section w e summarize sev eral useful results - for more details, see [9] or [13]. Ev ery rigid motion (orientation-preserving isometry) of the Euclidean space R n can b e written as the composition of a translation and a rigid motion that fixes the origin O . Moreo v er, every rigid motion that fixes the origin restricts to a rigid motion of the unit sphere S n − 1 = S 1 ( O ) ⊂ R n . The group of rigid motions of the unit sphere S n − 1 is the sp e cial ortho g- onal Lie group S O ( n ), and the group of rigid motions of R n is the sp e cial Euclide an group S E ( n ) ≃ S O ( n ) ⋉ R n , 3 Inciden tally , that was problem B1 on the 2008 Putnam Comp etition. HYPERSPHERE-BASED RESTRICTING CONDITIONS 7 a semidirect pro duct of S O ( n ) and the Ab elian group R n . The standard left action of S E ( n ) on R n , Φ : S E ( n ) × R n → R n , ( a, p ) 7− → a · p := a ( p ) , is smo oth and transitive. F or ev ery p oin t p ∈ R n , the stabilizer of p is the closed Lie subgroup G = Stab( p ) = { a ∈ S E ( n ) | a · p = p } ≃ S O ( n ) . F or every sphere S = S R ( p ) centered at p , the action of S E ( n ) on R n induces a smo oth and transitive action G × S → S. Lemma 3.1. L et q ∈ S and define ϕ : G → S, ϕ ( a ) = a · q . Then ϕ is c ontinuous, op en, and surje ctive. Pr o of. The G -orbit of q is ϕ ( G ) = Orbit G ( q ) = { a · q | a ∈ G } = S, so ϕ is surjective. The stabilizer of q in G is ϕ − 1 ( q ) = Stab G ( q ) = { a ∈ G | a · q = q } ≃ S O ( n − 1) . The natural map G/ Stab G ( q ) − → Orbit G ( q ) = S, [ a ] 7− → a · q , is a diffeomorphism. Under this identification, ϕ coincides with the quotien t pro jection G → G/ Stab G ( q ). Hence ϕ is contin uous, op en, and surjective. □ Definition 3.2. A property P of sets in R n is a rigid ge ometric pr op erty if for every set Γ ⊂ R n that has prop ert y P and rigid motion a ∈ S E ( n ), an y set that contains a · Γ has prop ert y P . The rigid geometric prop erties P that we will consider will b e based on simplic es , and we in tro duce some terminology at this p oin t - see [6] for details. Definition 3.3. An m − simplex σ = σ m in R n ( n ⩾ m ) is the con v ex h ull of m + 1 affinely indep enden t p oin ts, whic h are called the vertic es of σ m . An n − simplex in R n is called a ful l-dimensional simplex . F or example, a 1-simplex is a segmen t and a 2-simplex is a (solid) triangle. A triangle is a full-dimensional simplex in R 2 , but not in R 3 . The notions of equilateral, isosceles, and righ t triangles extend to higher dimensional simplices. A simplex is: (1) a r e gular simplex (or e quilater al ) if the Euclidean distances b et w een an y tw o distinct v ertices are all equal. 8 GABRIEL ISTRA TE AND CA T ALIN ZARA (2) an isosc eles simplex if there exists a vertex p such that all the dis- tances from p to the other v ertices of the simplex are equal - such a v ertex p is called an ap ex of the isosceles simplex. (3) a right simplex if there exists a vertex p such that the lines joining p with the other vertices are m utually orthogonal - such a v ertex p is called an ap ex of the righ t simplex. F or ev ery full-dimensional simplex σ n in R n , there exists a unique sphere S R ( p ) ⊂ R n that contains all the vertices of the simplex. That sphere is called the cir cumspher e , the center p is the cir cumc enter , and the radius R is the cir cumr adius . F or full-dimensional regular simplices and for right sim- plices in the tw o-dimensional space, the circumcenter is within the simplex, but that need not be the case for isosceles simplices, and is not the case for righ t simplices in dimensions greater that tw o. The prop ert y of a simplex in R n to b e equilateral, isosceles, or right are in v ariant under rigid motions of R n . An m − simplex σ in R n has a nonzero m − v olume v m ( σ ), and the m − v olume is also in v arian t under rigid motions of R n . Examples of rigid geometric prop erties that w e will consider include: P ( I S m ) : A set in R n has this prop ert y if it contains the vertices of an isosceles m − simplex. P ( E S m ) : A set in R n has this prop ert y if it contains the vertices of a regular m − simplex. P ( RS m ) : A set in R n has this prop ert y if it contains the vertices of a right m − simplex. P (v m , V ) : A set in R n has this prop ert y if it con tains the vertices of an m − simplex with m − volume in a fixed set V of admissible v alues. P k L N : A set in R n has this prop ert y if it con tains the vertices of an n − simplex with at least k of the N =  n +1 2  edge lengths in a fixed set L of admissible v alues. 4. Somewhere Comea ger Colorings In the next sections we study colorings satisfying conditions Q based on prop erties P inv olving simplices of sp ecific shap es. Ceder [3] sho wed that the plane can b e decomposed in to countably many sets, none of whic h con tains the v ertices of an equilateral triangle. Assuming HYPERSPHERE-BASED RESTRICTING CONDITIONS 9 the Con tin uum Hyp othesis, Da vies [4] ( n = 2), and Kunen [8] (all n ), con- structed partitions of R n in to countably many sets, none of which con tains the vertices of an isosceles triangle. Erd˝ os and Komj´ ath [5] obtained the same conclusion for n = 2 under Martin’s Axiom, and Sc hmerl [11] prov ed the same result for all n , without additional set-theoretic hypotheses. Erd˝ os and Komj´ ath [5], with a pro of completed in Bursics–Komj´ ath [2] and an alternativ e approach given in [12], sho w ed that assuming the Con tin uum Hyp othesis, the plane can b e colored with countably many colors so that no color class contains the v ertices of a righ t triangle. In all of these cases, coloring eac h set of the decomp osition with a different color yields colorings using countably man y colors that admit no monochro- matic triangles of the sp ecified type. F or eac h suc h coloring, the corresp ond- ing conditions Q are logically satisfied. How ev er, since every nonempt y open ball contains isosceles, equilateral, and righ t triangles, no mono c hromatic op en ball can exist. Th us, without additional regularity assumptions on the color classes, these conditions cannot imply the existence of a mono c hro- matic nonempty open ball. The regularity condition that we imp ose on colorings of the space R n is that they b e somewher e c ome ager c olorings - see Definition 4.1 b elo w. F or self-con tainmen t, w e include some basic definitions, prop erties, and results related to Baire Category here - for more details, see, for example, [10], [7]. A top ological space X is se c ond-c ountable if it has a countable basis - this will b e the case in our applications. A set A ⊂ X is nowher e-dense if int ( A ) = ∅ - that is, if the interior of the closure is empty . A set is me ager if it can b e written as the countable union of no where-dense sets, and otherwise is nonme ager . A set A is c ome ager if its complemen t X \ A is meager. A set A has the Bair e pr op erty if there is an op en set U suc h that the symmetric difference M = A ∆ U is meager - equiv alently , if A = M ∆ U is the symmetric difference of a meager set and an op en set. A nonempty subset U ⊂ X is a topological space with the induced top ol- ogy . A set A ⊂ X is c ome ager in U if A ∩ U is comeager in U . In particular, a set S ⊂ (0 , ∞ ) is c ome ager ne ar 0 if there exists ϵ > 0 such that S is comeager in (0 , ϵ ). A set A ⊂ X is somewher e c ome ager if there exists a nonempty op en set U ⊂ X suc h that A is comeager in U . If A ⊂ R n has the Baire prop ert y , then A is nonmeager if and only if A is somewhere comeager. Definition 4.1. A coloring f : R n → C is called a somewher e c ome ager c oloring if there exists a color c ∈ C such that the color class f − 1 ( c ) is somewhere comeager in R n . Prop osition 4.2. If f : R n → N is a c oloring such that al l c olor classes have the Bair e pr op erty, then f is a somewher e c ome ager c oloring. Pr o of. Since N is countable and R n is nonmeager, there exists a color c such that the color class f − 1 ( c ) is nonmeager. Since f − 1 ( c ) is nonmeager and has the Baire prop ert y , it follows that f − 1 ( c ) is somewhere comeager. □ 10 GABRIEL ISTRA TE AND CA T ALIN ZARA W e will use the following consequence of the Kuratowski-Ulam theorem. Lemma 4.3. L et p ∈ R n and R > 0 . If A ⊂ B R ( p ) is c ome ager in B R ( p ) , then the set G p = { r ∈ (0 , R ) | A is c ome ager in S r ( p ) 4 } is c ome ager in (0 , R ) . In p articular, G p is infinite. Pr o of. Consider the punctured op en ball D = { p + r q | 0 < r < R, q ∈ S n − 1 } . Let Φ : (0 , R ) × S n − 1 → D , defined by Φ( r , q ) = p + r q . Then Φ is a homeomorphism, and since A \ { p } is comeager in D , its inv erse image Φ − 1 ( A \ { p } ) is comeager in (0 , R ) × S n − 1 . By the Kuratowski-Ulam theorem, the set { r ∈ (0 , R ) | { q | ( r, q ) ∈ Φ − 1 ( A \ { p }} is comeager in S n − 1 } is comeager in (0 , R ). Under the homeomorphism Φ, this set corresp onds exactly to G p , which completes the pro of. □ F or the remainder of this pap er we consider colorings f : R n → C ⊂ R that are somewher e c ome ager , and conditions Q for which all p oin ts of the space are admissible cen ters (Ω = R n ) and for whic h the set R of admissible radii is c ome ager ne ar 0. Remark 4.4. T o streamline notation, we write Q ∗ P , R ϵ = Q C , P , R n , R to in- dicate that the ab o v e conditions are satisfied. Accordingly , the statement that f satisfies Q ∗ P , R ϵ means that: (1) f : R n → C is a somewhere comeager coloring. (2) f satisfies Q C , P , R n , R with R comeager in (0 , ϵ ). Definition 4.5. A prop ert y P is a uniform-c ap pr op erty if there exists δ ∈ (0 , 1) such that for every admissible sphere S = S r ( p ) and every x ∈ S , the op en cap B δ r √ 2 ( x ) ∩ S r ( p ) has the prop ert y P . A prop ert y P is a c ountable uniform-c ap pr op erty if the op en cap B δ r √ 2 ( x ) ∩ S r ( p ) con tains a coun table subset that has prop ert y P . The definition ab o v e states that a prop ert y P is a uniform-cap prop ert y if every open hemisphere B r √ 2 ( x ) ∩ S r ( p ) of an admissible sphere S r ( p ) con tains a proper cap - of radius δ r √ 2 - having prop ert y P , and the relativ e size (radius) of that cap is uniformly b ounded on all admissible spheres. The main result of this section is the follo wing theorem. Theorem 4.6. L et f b e a c oloring that satisfies Q ∗ P , R ϵ . (1) If every c ome ager subset of any admissible spher e has pr op erty P , then f is somewher e lo c al ly mono chr omatic. 4 for the induced top ology on S r ( p ). HYPERSPHERE-BASED RESTRICTING CONDITIONS 11 (2) If f is somewher e lo c al ly mono chr omatic and P is a uniform-c ap pr op erty, then f is mono chr omatic. Pr o of. (1) Since f is somewhere comeager, there exist a color c ∈ C and a nonempt y op en set U ⊂ R n suc h that f − 1 ( c ) is comeager in U . Let p ∈ U and R > 0 suc h that B R ( p ) ⊂ U . Then f − 1 ( c ) is comeager in B R ( p ) and from Lemma 4.3 we conclude that the set G p = { r ∈ (0 , R ) | f − 1 ( c ) is comeager in S r ( p ) } is comeager in (0 , R ). Let δ = min( R , ϵ ). Both G p and R are comeager in (0 , δ ), so their inter- section, A p = { r ∈ (0 , δ ) ∩ R | f − 1 ( c ) is comeager in S r ( p ) } is also comeager in (0 , δ ), hence nonempty . Thus, there exists an admissi- ble radius r suc h that the set f − 1 ( c ) is comeager in S r ( p ), an admissible sphere. By h yp othesis, this comeager set has prop ert y P , and since f sat- isfies Q ∗ P , R ϵ , it follows that f ( p ) = c . As this holds for every p ∈ U , the coloring f is constant on the op en set U , and hence f is somewhere lo cally mono c hromatic. (2) Supp ose that f is somewhere lo cally mono c hromatic. Then there exists a mono c hromatic op en nonempt y ball B r 0 ( p ). Let R = sup { r > 0 | B r ( p ) is mono c hromatic } . Clearly R ⩾ r 0 > 0. Assume, for contr adiction, that R < ∞ . Let q ∈ S R ( p ). F or a small enough admissible radius r > 0, the in tersection S r ( q ) ∩ B R ( p ) is as close to a hemisphere on S r ( q ) as w e wan t - in particular, we can c ho ose r so that the in tersection con tains an op en cap of radius 1+ δ 2 r √ 2. Keeping r fixed and mo ving q sligh tly a w a y from p w ould yield a bit smaller in tersection S r ( q ) ∩ B R ( p ), but w e can still k eep it as close to a hemisphere on S r ( q ) as w e wan t - to contain an open cap of radius δ r √ 2 - by staying close enough, d ( p, q ) < R + t 0 for some suitably small t 0 > 0. Then S r ( q ) has a monochromatic op en cap of radius δ r √ 2, and that cap has property P , therefore, f ( q ) = f ( p ). W e conclude that B R + t 0 ( p ) is mono c hromatic, con tradicting the definition of R as the supremum. Therefore, R = ∞ , and f is mono c hromatic on R n . □ Theorem 4.6 has the follo wing consequence for rigid geometric prop erties. Theorem 4.7. L et f b e a c oloring that satisfies Q ∗ P , R ϵ for a rigid ge ometric pr op erty P . (1) If every admissible spher e c ontains a c ountable subset that has pr op- erty P , then f is somewher e lo c al ly mono chr omatic. (2) If P is a c ountable uniform-c ap pr op erty, then f is mono chr omatic. Pr o of. Let S = S R ( p ) b e an admissible sphere and let A ⊂ S b e any comea- ger subset. Let Γ = { q i } i ∈ I b e a countable subset of S that has prop ert y 12 GABRIEL ISTRA TE AND CA T ALIN ZARA P . F or i ∈ I let ϕ i : G → S , ϕ i ( a ) = a · q i , as in Lemma 3.1. Since ϕ i is con tin uous, op en, and surjectiv e, and A is comeager in S , it follo ws that ϕ − 1 i ( A ) is comeager in G , for all i ∈ I . As I is coun table, the intersection of these sets is comeager, hence, there exists a ∈ \ i ∈ I ϕ − 1 i ( A ) ⊂ G, and then a · Γ ⊂ A . Since P is a rigid geometric property , A has prop ert y P . By the first part of Theorem 4.6, f is somewhere lo cally mono c hromatic. The h yp othesis of the second statemen t implies the condition in the first one, hence f is somewhere lo cally mono c hromatic. Since the group of rigid motions of a sphere acts transitively and P is a rigid geometric prop ert y , those hypotheses also imply that the condition of the second part of Theo- rem 4.6 are satisfied. Therefore, f is mono c hromatic. □ Corollary 4.8. L et f b e a c oloring that satisfies Q ∗ P , R ϵ for a rigid ge ometric pr op erty P . If every op en set of every spher e has a c ountable subset with pr op erty P , then f is mono chr omatic. 5. Simplex Shape Conditions In this section w e study colorings of R n satisfying Q ∗ P , R ϵ , where P is one of the rigid geometric prop erties P ( I S m ) , P ( E S m ) , or P ( RS m ) , with 2 ⩽ m ⩽ n . The main result of this section is the follo wing. Theorem 5.1. L et f b e a c oloring or R n that satisfies a c ondition Q ∗ P , R ϵ , wher e P is one of P ( I S m ) , P ( E S m ) , or P ( RS m ) , with 2 ⩽ m ⩽ n . (1) If m < n , or m = n and P = P ( I S n ) , or m = n ⩾ 3 and P = P ( RS n ) , then f is mono chr omatic. (2) If m = n and P = P ( E S n ) , or m = n = 2 and P = P ( RS 2 ) , then f is somewher e mono chr omatic, but not ne c essarily mono chr omatic. Pr o of. Every sphere in R m con tains the vertices of isosceles, equilateral, and right m − simplices. Since every sphere in R n con tains a subset isomet- ric with a sphere in R m , it follo ws that every sphere in R n con tains the v ertices of isosceles, equilateral, and righ t m − simplices. By the first part of Theorem 4.7, the coloring f is somewhere mono c hromatic. Supp ose m < n . Let S b e a sphere in R n , let U ⊂ S b e a nonempty op en subset of S , and let p ∈ U . Then there exists an op en ball B r ( p ) in R n suc h that p ∈ B r ( p ) ∩ S ⊂ U . The intersection S r/ 2 ( p ) ∩ S is isometric with a sphere in R n − 1 and, since m ⩽ n − 1, it contains a subset isometric with a sphere S ′ in R m . The sphere S ′ con tains the vertices of isosceles, equilateral, and right m − simplices, and therefore, U con tains the vertices of isosceles, equilateral, and righ t m − simplices. By Corollary 4.8, the coloring f is mono c hromatic. HYPERSPHERE-BASED RESTRICTING CONDITIONS 13 Supp ose that m = n and P = P ( I S n ) . Let U , p , and r be as ab o v e, and let σ ′ b e any ( n − 1) − simplex with v ertex set V ⊂ S r/ 2 ( p ) ∩ S . Then the n − simplex with vertex set V ∪ { p } is an isosceles n − simplex with vertices in U . By Corollary 4.8, the coloring f is mono c hromatic. Supp ose that m = n ⩾ 3 and P = P ( RS n ) . Let p 0 , p 1 , . . . , p n b e the v ertices of a right simplex σ with ap ex p 0 and s u ch that d ( p 0 , p i ) = a > 0 for all i = 1 , . . . , n . The circumradius of σ is r = a √ n/ 2. Therefore, if 1 > δ > r 2 n , then ev ery op en cap of radius δ r √ 2 of ev ery sphere of radius r contains a finite set with prop ert y P ( RS n ) . By the second part of Theorem 4.7, the coloring f is mono c hromatic. Finally , we construct a (counter)example to sho w that if m = n and P = P ( E S n ) , or if m = n = 2 and P = P ( RS 2 ) , then f do es not ha ve to b e mono c hromatic. Define a coloring f : R n → Z b y horizontal strips: f ( x 1 , . . . , x n ) = ⌊ x n ⌋ ∈ Z . The color class of c ∈ Z is f − 1 ( c ) = R n − 1 × [ c, c + 1) . Eac h f − 1 ( c ) is Borel, hence f is a Baire coloring. Moreov er, each color class is conv ex. F or equilateral simplices, and for full-dimensional right simplices in R 2 (righ t-angle triangles), the circumcenter lies in the conv ex h ull of the ver- tices. Since each color class is conv ex, if the v ertices of an y equilateral or righ t simplex lie in the same color class, then so does the circumcenter. Th us f is a Baire coloring that satisfies Q ∗ ( E S n ) , R ϵ and, for n = 2, Q ∗ ( RS 2 ) , R ϵ . Ho w ever, f is not mono c hromatic. The construction ab o v e can easily b e modified to the case of finitely many colors. F or |C | ⩾ 2, finite, merge all color classes with c < 0 into one class and all color classes with c ⩾ |C | − 2 into another, leaving the remaining classes unc hanged. This yields a Baire coloring with |C | colors that satisfies b oth Q ∗ ( E S n ) , R ϵ and Q ∗ ( RS 2 ) , R ϵ , but is not mono c hromatic. □ 6. Edge-Length Conditions In this section w e study conditions based on simplices with restricted edge lengths. Let L ⊂ (0 , ∞ ) be a nonempt y set and 1 ⩽ k ⩽ N =  n +1 2  . A somewhere comeager coloring f : R n → C satisfies condition Q ∗ k L N , R ϵ if for every color c ∈ C and every admissible sphere S r ( p ), if f − 1 ( c ) ∩ S r ( p ) con tains the v ertices of an n − simplex σ with at least k edge-lengths in L , then f ( p ) = c . The following result characterizes the sets L for whic h ev ery coloring satisfying the condition Q ∗ ( k L N ) , R ϵ is mono c hromatic. 14 GABRIEL ISTRA TE AND CA T ALIN ZARA Theorem 6.1. L et 1 ⩽ k < N =  n +1 2  . The fol lowing ar e e quivalent: (i) Every c oloring f : R n → C satisfying Q ∗ ( k L N ) , R ϵ is mono chr omatic. (ii) inf ( L ) = 0 . Pr o of. W e first pro v e ii ) → i ), by showing that the rigid geometric prop ert y P (( N − 1) L N ) satisfies the hypothesis of Corollary 4.8 - then P ( k L N ) satisfies the same hypothesis for all 1 ⩽ k ⩽ N − 1. Let S = S 0 b e any sphere in R n (here 0 is the index, not the radius) and let U = U 0 ⊂ S 0 = S b e an op en subset. Let p 0 ∈ U 0 and t 0 ∈ (0 , ∞ ) such that B t 0 ( p 0 ) ∩ S 0 ⊂ U 0 . Since inf ( L ) = 0, there exists r 0 ∈ L ∩ (0 , t 0 ). Define S 1 = S r 0 ( p 0 ) ∩ S 0 ⊂ U 0 . Then d ( p 0 , q ) = r 0 ∈ L for all q ∈ S 1 and S 1 is isometric to a sphere of radius t 1 > 0 in R n − 1 . Let p 1 ∈ S 1 . W e rep eat the construction ab o v e for ( p 1 , S 1 , U 1 = S 1 , t 1 ) instead of ( p 0 , S 0 , U 0 , t 0 ), and then iterate. W e construct p oin ts p 0 , p 1 , . . . , p n − 2 on a decreasing chain of spheres S 0 ⊃ S 1 ⊃ · · · ⊃ S n − 2 , suc h that the dimen- sions of the spheres decrease b y 1 at eac h step, and d ( p i , p j ) = r i ∈ L for all 0 ⩽ i < j ⩽ n − 2. W e can rep eat the construction one more time, using circles S n − 2 and S r n − 2 ( p n − 2 ), and let S n − 1 = { p n − 1 , p n } b e the resulting in tersection. Then the n − simplex with v ertex set Γ = { p 0 , p 1 , . . . , p n − 1 , p n } ⊂ U has the edge-lengths, with the p ossible exception of d ( p n − 1 , p n ), in the set L of admissible edge-lengths. Hence Γ ⊂ U has property P (( N − 1) L N ) . By Corollary 4.8, for ev ery 1 ⩽ k < N , ev ery coloring f : R n → C satisfying Q ∗ ( k L N ) , R ϵ is mono c hromatic. Next we prov e, b y r e ductio ad absur dum , that i ) → ii ). Assume that i ) holds but inf ( L ) > 0. Cho ose δ > 0 such that δ √ n < inf ( L ), and partition R n in to half-op en h yp ercubes Q ( x 1 ,x 2 ,...,x n ) = n × j =1 [ δ x j , δ ( x j + 1)) , ( x 1 , . . . , x n ) ∈ Z n . Fix a bijection κ : Z n → N , and assign to each p oin t p ∈ Q ( x 1 ,...,x n ) the color f ( p ) = κ ( x 1 , . . . , x n ). Clearly f is a somewhere comeager coloring. An y segmen t joining points of the same color has length smaller than δ √ n < inf ( L ). Hence, if an n − simplex has an edge of length in L , then it can not ha v e all the vertices of the same color. Then, for all 1 ⩽ k < N , the coloring f satisfies Q ∗ ( k L N ) , R ϵ , but is not monochromatic, contradicting i ). Hence, inf ( L ) = 0. □ The case k = N =  n +1 2  exhibits a qualitativ ely differen t b eha vior, as sho wn in the follo wing example. Example 6.2. Let L ∈ (0 , ∞ ) b e a countable set with inf ( L ) = 0 - for example, the set of v alues of a decreasing sequence con verging to 0. The cardinalit y of the set E n = { ( i, j ) | 0 ⩽ i < j ⩽ n } HYPERSPHERE-BASED RESTRICTING CONDITIONS 15 is | E n | = N =  n +1 2  and the cardinalit y of the set of functions h : E n 7→ L is ℵ N 0 = ℵ 0 . W e call suc h a function h fe asible if there exists an n − simplex in R n , with vertex set { p 0 , p 1 , . . . , p n } , such that for all ( i, j ) ∈ E n w e hav e d ( p i , p j ) = h ( i, j ). Not all functions h are feasible - for example, some violate the triangle inequality . F or a feasible h , there exists a unique - up to isometries - n − simplex σ in R n with those prescrib ed edge lengths. Therefore, the set of circumradii of simplices realizing the feasible functions h is countable. Let R b e the complement in (0 , ∞ ) of that countable set - then R is comeager, hence, comeager near 0. No admissible sphere contains the ver- tices of an n − simplex with all edge-lengths in L . Therefore, ev ery coloring f : R n → C that is somewhere comeager satisfies condition Q ∗ ( N L N ) , R ϵ , hence that condition do es not force a coloring to b e mono c hromatic, and not ev en somewhere lo cally mono c hromatic. Th us, for ( N L N ), a sufficient condition for mono c hromaticit y must be stronger than inf ( L ) = 0. W e give suc h a condition b elo w. Theorem 6.3. If the set L of admissible e dge-lengths is c ome ager ne ar 0, then every c oloring f : R n → C satisfying Q ∗ ( N L N ) , R ϵ is mono chr omatic. Pr o of. Let δ > 0 such that L is comeager in (0 , δ ). Let S b e a sphere of radius R > 0 in R n and U ⊂ S an op en subset of S . Let p 0 ∈ U and t 0 > 0 suc h that p 0 ∈ S ∩ B t 0 ( p 0 ) ⊂ U . Let h : (0 , t 0 ) → (0 , ∞ ) b e the function defined as follo ws. F or t ∈ (0 , t 0 ), the intersection S ∩ S t ( p 0 ) is isometric with a sphere S ′ ⊂ R n − 1 . Let h ( t ) b e the length of the edges of a regular ( n − 1) − simplex inscrib ed in S ′ . F or small enough t 0 > 0, the con tinuous function h : (0 , t 0 ) → (0 , ∞ ) is increasing. Let r 0 ∈ (0 , t 0 ) such that max( r 0 , h ( r 0 )) < δ and h : (0 , r 0 ) → (0 , h ( r 0 )) is a homeomorphism. Since both L and h ( L ∩ (0 , r 0 )) are comeager in (0 , h ( r 0 )), it follo ws that their intersection is nonempty . Let r ∈ L ∩ (0 , r 0 ) suc h that h ( r ) ∈ L . Let { p 1 , . . . , p n } b e the set of vertices of a regular ( n − 1) − simplex in- scrib ed in the sphere S ∩ S r ( p 0 ) ⊂ U . Then the n − simplex with set of v ertices { p 0 , p 1 , . . . , p n − 1 , p n } ⊂ U has the edge-lengths r, h ( r ) ∈ L . By Corollary 4.8, every coloring f : R n → C satisfying Q ∗ ( N L N ) , R ϵ is mono c hro- matic. □ 7. V olume Conditions In this section w e study conditions based on simplices with restricted admissible volumes. Let V ⊂ (0 , ∞ ) b e a nonempty set and let 2 ⩽ m ⩽ n . A somewhere comeager coloring f : R n → C satisfies condition Q ∗ (v m , V ) , R ϵ if for every color c ∈ C and ev ery admissible sphere S r ( p ), if f − 1 ( c ) ∩ S r ( p ) contains the v ertices of an m − simplex σ with v m ( σ ) ∈ V , then f ( p ) = c . 16 GABRIEL ISTRA TE AND CA T ALIN ZARA The follo wing result c haracterizes the sets V for whic h ev ery coloring satisfying the condition Q ∗ (v m , V ) , R ϵ is mono c hromatic. Theorem 7.1. L et 2 ⩽ m ⩽ n . The fol lowing ar e e quivalent: (i) Every c oloring f : R n → C satisfying Q ∗ (v m , V ) , R ϵ is mono chr omatic. (ii) inf ( V ) = 0 . Pr o of. W e first pro v e ii ) → i ), by showing that the rigid geometric prop ert y P (v m , V ) satisfies the hypothesis of the second part of Theorem 4.7. Let S b e a sphere of radius R > 0 in R n and U ⊂ S an op en subset of S . Let p 0 ∈ U and r 0 > 0 such that p 0 ∈ S ∩ B r 0 ( p 0 ) ⊂ U . F or r ∈ (0 , r 0 ), let { p 1 , p 2 , . . . , p n } b e the vertices of a regular ( n − 1) − simplex inscrib ed in S ∩ S r ( p 0 ). Let ∆ = ∆( r ) be the m − simplex with vertices { p 0 , p 1 , . . . , p m } , and let h : (0 , r 0 ) → (0 , ∞ ), defined b y h ( r ) = v m (∆( r )). Fix r 1 ∈ (0 , r 0 ). Since inf ( V ) = 0, there exists v ∈ V suc h that 0 < v < h ( r 1 ). Since h is con tin uous and lim r → 0 h ( r ) = 0, there exists r ∈ (0 , r 1 ) such that h ( r ) = v . Then the m − simplex ∆( r ) has all vertices in U and has m − v olume in V . By Corollary 4.8, every coloring satisfying Q ∗ (v m , V ) , R ϵ is mono c hromatic. Next we prov e, b y r e ductio ad absur dum , that i ) → ii ). Assume that i ) holds but inf ( V ) > 0. W e extend the m − v olume function to con v ex h ulls of m + 1 p oin ts in R n b y setting the volume to be 0 if the points are not affinely independent. Let □ be the closed unit hypercub e in R n . Let h : □ m +1 → [0 , ∞ ] be defined b y h ( p 0 , . . . , p m ) = v m (∆ p 0 ,...,p m ). As h is con tin uous and □ m +1 is compact, there exists c m > 0 such that h ( p 0 , . . . , p m ) ⩽ c m for all ( p 0 , . . . , p m ) ∈ □ m +1 . Then, for every δ > 0, closed h yp ercube □ δ ⊂ R n of edge-length δ , and p oin ts p 0 , . . . , p m ∈ □ δ , we ha v e v m ( p 0 , . . . , p m ) ⩽ c m δ m . Fix δ > 0 such that c m δ m < inf ( V ). The coloring constructed in the second half of the pro of of Theorem 6.1 satisfies the condition Q ∗ (v m , V ) , R ϵ but is not mono c hromatic, contradicting i ). 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F acul ty of Ma thema tics and Computer Science, University of Bucharest, Bucharest, R omania Email address : gabriel.istrate@unibuc.ro Dep ar tment of Ma thema tics, University of Massachusetts Boston, Boston, MA, United St a tes of America Email address : catalin.zara@umb.edu