Symmetric monochromatic subsets in colorings of the Lobachevsky plane

SYMMETRIC MONOCH R OMA TIC SUBSETS IN COLORINGS OF THE LOBA CHEVSKY PLANE T ARAS BANAKH, AR TEM DU D KO, AND DU ˇ SAN REPOV ˇ S Abstract. W e pr o ve that for eac h partition of the Lobac hevsky plane i n to finitely many Borel pieces one of the cells of the partition contains an un- bounded cen trally symmetric subset. It fo llows from [B 1 ] (see also [B P 1 , Theorem 1]) that for each partition of the n -dimensional space R n int o n pieces one of the pieces contains an unbounded centrally symmetric s ubs et. O n the other hand, R n admits a par tition into ( n + 1) Borel pieces cont aining no un bo unded centrally sy mmetr ic subset. F or n = 2 such a pa r tition is drawn at the picture: ❜ ❜ ❜ ❜ ❜ ✧ ✧ ✧ ✧ ✧ B 0 B 1 B 2 T aking the same partition of the Lobachevsky plane H 2 , we can see that each cell B i do es contain a unbounded ce n trally symmetric s ubs et (for such a set just take any h yper bo lic line lying in B i ). W e call a subset S of the h yp erb olic plane H 2 c entr al ly symmetric or else symmetric with r esp e ct to a p oint c ∈ H 2 if S = f c ( S ) where f c : H 2 → H 2 is the inv o lutive iso metr y of H 2 assigning to each point x ∈ H 2 the unique po int y ∈ H 2 such that c is the midp oint o f the segment [ x, y ]. The ma p f c is ca lle d the c entr al symmetry of H 2 with r esp ect to the p oint c . The following theorem sho ws tha t the Lobachevsky plane differs dramatica lly from the Euclidean plane fr om the Ramsey po int of view. Theorem 1. F or any p artition H 2 = B 1 ∪ · · · ∪ B m of t he L ob achevsky plane into finitely many Bor el pie c es one of t he pie c es c ontains an unb ounde d c en tr al ly symmetric subset. Pr o of. W e shall prove a bit mor e: given a par tition H 2 = B 1 ∪ · · · ∪ B m of the Lobachevsky plane int o m Borel pieces we shall find i ≤ m and an unbounded subset S ⊂ B i symmetric with resp ect to some p oint c in an a rbitrarily sma ll neighborho o d o f some finite set F ⊂ H 2 depe nding only on m . T o define this se t F it will b e conv enient to work in the Poincar´ e mo del of the Lobachevsky plane H 2 . In this mo del the hyperb olic plane H 2 is identified with 1991 Mathematics Subje ct Classific ation. 05D10, 51M09, 54H09. 1 2 T ARAS BANAKH, AR TEM DUDKO, AND DU ˇ SAN REP OV ˇ S the unit disk D = { z ∈ C : | z | < 1 } on the complex plane a nd h yp e r b olic lines are just segments of circles orthog onal to the b oundary of D . Let D = { z ∈ C : | z | ≤ 1 } be the hyperb olic plane D with attached ideal line. F or a rea l num b er R > 0 the set D R = { z ∈ C : | z | ≤ 1 − 1 /R } ca n be thought as a hyperb olic disk of increasing radius as R tends to ∞ . On the b ounda ry of the unit disk D consider the ( m + 1)-element set A = { z ∈ C : z m +1 = 1 } . F or a ny tw o distinct po int s x, y ∈ A by [ x | y ] ∈ D we denote the “E uclidean” midpo in t of the arc in D that connects the p oints x, y and lies on a hyper bo lic line in H 2 = D . Then F = { [ x | y ] : x, y ∈ A, x 6 = y } is a finite subset of cardinality | F | ≤ m ( m + 1) / 2 in the unit disk D . W e claim that for any op en neighbo rho o d W of F in C one of the cells of the partition H 2 = B 1 ∪ · · · ∪ B m contains an un b ounded subset symmetric with resp ect to some p o in t c ∈ W . T o derive a contradiction we assume the conv e r se and conclude that for every p oint c ∈ W a nd every i ≤ m the se t B i ∩ f c ( B i ) is bo unded in H 2 . F or every n ∈ N consider the s et C n = { c ∈ W : m [ i =1 B i ∩ f c ( B i ) ⊂ D n } . W e claim that C n is a coana lytic subse t of W . The latter mea ns that the complement W \ C n is analytic, i.e., is the co ntin uous image of a Polish spa ce. Observe that W \ C n = { c ∈ W : ∃ i ≤ m ∃ x ∈ D \ D n , x ∈ B i and x ∈ f c ( B i ) } = pr 2 ( E ) where pr 2 : D × D → D is the pro jection on the second factor and E = m [ i =1 { ( x, c ) ∈ D × W : x ∈ D \ D n , x ∈ B i and f c ( x ) ∈ B i } is a Borel s ubset of D × W . B eing a Borel subset of the Polish space D × W , the space E is ana lytic and so is its contin uous image pr 2 ( E ) = W \ C n . Then C n is coanalytic and hence has the Ba ire pr op erty [Ke, 2 1.6], which means that C n coincides with an op en subset U n of W mo dulo s ome meage r set. The latter means that the s ymmetric difference U n △ C n is meager (that is, o f the fir st Baire ca teg ory in W ). Since C n ⊂ C n +1 , we may assume that U n ⊂ U n +1 for all n ∈ N . Le t U = S ∞ n =1 U n and M = S ∞ n =1 U n △ C n . T aking in to a ccount tha t W = S ∞ n =1 C n , w e conclude that W \ U = ∞ [ n =1 C n \ ∞ [ n =1 U n ⊂ ∞ [ n =1 C n \ U n ⊂ ∞ [ n =1 C n △ U n = M which implies that the o p e n set U has meager complement and thus is dens e in W . W e claim that F ⊂ h − 1 ( U ) for some isometry h of the hyperb olic plane H 2 = D . F or this co nsider the natura l action µ : Is o( H 2 ) × D → D , µ : ( h, x ) 7→ h ( x ) of the iso metry group Iso( H 2 ) of the hyperb olic plane H 2 = D . It is easy to s ee that for every x ∈ D the map µ x : Iso( H 2 ) → D , µ x : h 7→ h ( x ), is con tinuous and ON COLORINGS OF THE LOBACHEVSKY PLANE 3 op en (with resp ect to the compact- op en topo logy on Iso( H 2 )). It follows that the set \ x ∈ F µ − 1 x ( W ) = { h ∈ Is o( H 2 ) : f ( F ) ⊂ W } is an op en neighborho o d o f the neutral element of the gro up Iso( H 2 ). T aking int o account that U is o p en and dense in W a nd for every x ∈ F the map µ x : Iso( H 2 ) → D is op en, we conclude that the preima ge the set µ − 1 x ( U ) is op en and dense in µ − 1 x ( W ) ⊂ Iso( H 2 ). Then the intersection T x ∈ F µ − 1 x ( U ), b eing an op en dense subset o f T x ∈ F µ − 1 x ( W ), is not empty and hence contains some isometry h having the des ired pr op erty: F ⊂ h − 1 ( U ). Since F is finite, there is R ∈ N with F ⊂ h − 1 ( U R ). F or a c o mplex num b er r ∈ D consider the set r A = { r z : z ∈ A } ⊂ D and let F r = { [ x | y ] : x, y ∈ r A, x 6 = y } ⊂ D , where [ x | y ] s tands for the midpo int of the hyperb o lic segment co nnecting x a nd y in H 2 . It c an b e shown that for any distinct po int s x, y ∈ A the “hyperb olic ” midpo in t [ rx | r y ] tends to the “Euclidea n” midp oint [ x | y ] as r tends to 1 . Suc h a contin uity yields a neighborho od O 1 of 1 such that F r ⊂ h − 1 ( U R ) for all r ∈ O 1 ∩ D . It is clea r that for an y p oints x, y ∈ A the map f x,y : D → D , f x,y : r 7→ [ r x | ry ] is o p e n and contin uous. Consequently , the preima ge f − 1 x,y ( h − 1 ( M )) is a meager subset o f D and so is the union M ′ = S x,y ∈ A f − 1 x,y ( h − 1 ( M )). So, we ca n find a non- zero p oint r ∈ O 1 \ M ′ so close to 1 that the set rA is disjoint with the hyp e rb olic disk h − 1 ( D R ). F or this point r w e sha ll get F r ∩ h − 1 ( M ) = ∅ . The set r A co nsists of m + 1 po ints. Consequently , so me cell h − 1 ( B i ) of the partition D = h − 1 ( B 1 ) ∪ · · · ∪ h − 1 ( B m ) c ontains tw o distinct po int s rx, r y of rA . Those points are symmetric with res pe ct to the po in t [ rx | r y ] ∈ F r ⊂ h − 1 ( U R ) \ h − 1 ( M ) . Then the imag es a = h ( rx ) and b = h ( r y ) b elong to B i and are s ymmetric with resp ect to the p oint c = h ([ r x | ry ]) ∈ U R \ M ⊂ C R . It follows from the definition of C R that { a , b } ⊂ B i ∩ f c ( B i ) ⊂ D R , which is not the case b ecause rx, r y / ∈ h − 1 ( D R ).  W e do not k now if Theo rem 1 is true for any finite (not nec e ssarily Borel) pa rti- tion of the Lobachevsky plane H 2 . F or partitions of H 2 int o tw o pieces the Borel assumption is sup efluous. Theorem 2. Ther e is a subset T ⊂ H 2 of c ar dinality | T | = 3 such that for any p artition H 2 = A 1 ∪ A 2 of H 2 into two pie c es either A 1 or A 2 c ontains an u n b ounde d subset, symmetric with r esp e ct to some p oint c ∈ T . Pr o of. L e mma 1 b elow allows us to find an equilatera l tr ia ngle △ c 0 c 1 c 2 on the Lobachevsky plane H 2 such that the comp osition f c 2 ◦ f c 1 ◦ f c 0 of the symmetries with r esp ect to the p oints c 0 , c 1 , c 2 coincides with the rotation on the a ngle 2 π / 3 around so me p oint o ∈ H 2 . Consequently ( f c 2 ◦ f c 1 ◦ f c 0 ) 3 is the identit y isometr y of H 2 . W e cla im that for any partition H 2 = A 1 ⊔ A 2 of the Lobachevsky plane into t wo pieces one of the pieces cont ains an unbo unded subset sy mmetric with resp ect 4 T ARAS BANAKH, AR TEM DUDKO, AND DU ˇ SAN REP OV ˇ S to so me p oint in the triangle T = { c 0 , c 1 , c 2 } . Assuming the conv erse, we conclude that the set B = [ c ∈ T 2 [ i =1 A i ∩ f c ( A i ) is bounded. It follows that tw o points x, y ∈ H 2 \ B , symmetric w ith r esp ect to a center c ∈ T cannot b elo ng to the same cell A i of the partition. Given a p oint x 0 ∈ H 2 consider the sequence o f points x 1 , . . . x 9 defined by the recursive formula: x i +1 = f c i mod 3 ( x i ). It follo ws that x 9 = ( f c 2 ◦ f c 1 ◦ f c 0 ) 3 ( x 0 ) = x 0 . T aking x 0 sufficiently far from the center o of r otation we can g uarantee that none of the p oints x 0 , . . . , x 9 belo ngs to B . The p oint x 0 belo ngs either to A 1 or to A 2 . W e lose no generality as suming that x 0 ∈ A 2 . Since the po int s x 0 , x 1 / ∈ B are symmetric with resp ect to c 0 and x 0 ∈ A 2 , we get that x 1 ∈ H 2 \ A 2 = A 1 . By the sa me reason x 1 , x 2 cannot simult aneously belo ng to A 1 and hence x 2 ∈ A 2 . Contin uing in this fashion w e conclude that x i belo ngs to A 1 for o dd i and to A 2 for even i . In particular, x 9 ∈ A 1 , which is not po ssible b eca use x 9 = x 0 ∈ A 2 .  Lemma 1. Ther e is an e quilater al triangle △ AB C on the L ob achevsky plane su ch that the c omp osition f C ◦ f B ◦ f A of the symmetries with r esp e ct to t he p oints A, B , C c oincides with the r otation on the angle 2 π / 3 ar oun d some p oint O . Pr o of. F or a po sitive real num b er t cons ider a n equilateral triangle △ AB C with side t the on the Lo bachevsky plane. Let M b e the midp oint o f the s ide AB and l be the line through C that is orthog onal to the line C M . Co nsider also the line p that is or thogonal to the line AB a nd pass e s thro ugh the p oint P s uch that A is the midpo in t b etw een P and M . Observe that | P M | = | AB | = t and for sufficiently small t the lines p and l intersect at some point O . s C l s A s B s M s X ′ s P p s X s O It is easy to see that the compo sition f B ◦ f A is the shift along the line AB on the distance 2 t and hence the image f B ◦ f A ( O ) of the point O is the point sy mmetr ic to O with resp ect to the po int C . Co nsequently , f C ◦ f B ◦ f A ( O ) = O , which mea ns ON COLORINGS OF THE LOBACHEVSKY PLANE 5 that the isometry f C ◦ f B ◦ f A is a rotation of the Lobachevsky plane a round the po int O on s ome a ngle ϕ t . T o estimate this angle, consider the p oint X s uch tha t P is the midp oint be tw een X and M . Then | X M | = 2 t and consequently , f B ◦ f A ( X ) = M while X ′ = f C ◦ f B ◦ f A = f C ( M ) is the p oint on the line C M such that C is the midpoint betw een X ′ and M . It follows that | X ′ X | ≤ | X M | + | M X ′ | < 2 t + 2 t = 4 t . Observe that for small t the p oint X ′ is near to the p oint, symmetric to X with resp ect to O , which means that the ang le ϕ t = ∠ X O X ′ is close to π for t c lo se to zero. On the other hand, for very large t the lines p and l on the Lobachevsky plane do not intersect. So we can consider the smallest upp er b ound t 0 of num ber s t for which the lines l and p meet. F or v alues t < t 0 near to t 0 the p oint O tends to infinit y as t tends to t 0 . Since the leng th of the side X X ′ of the triang le △ X OX ′ is bo unded b y 4 t 0 the a ngle ϕ t = ∠ X OX ′ tends to zero as O tends to infinity . Since the angle ϕ t depe nds contin uous on t and decr eases from π to zero as t incre ases from zero to t 0 , there is a v alue t such that ϕ t = 2 π / 3. F o r such t the comp o sition f C ◦ f B ◦ f A is the r otation ar ound O on the angle 2 π / 3.  Some comments and Open Problems In contrast to Theor em 1, Theore m 2 is true for the Euclidean plane E 2 even in a strong er form: for any s ubs e t C ⊂ E 2 not lying on a line and any partition E 2 = A 1 ∪ A 2 one o f the cells o f the pa rtition contains an un bo unded subset symmetric w ith resp ect to some center c ∈ C , see [B 2 ]. Having in mind this result let us c all a subset C of a Lobachevsky or E uclidean space X c ent r al for (Bor el) k -p artitions if for any partition X = A 1 ∪ · · · ∪ A k of X int o k (Bor el) pieces o ne of the pie c es contains an unbounded mono chromatic subset S ⊂ X , symmetric with resp ect to so me point c ∈ C . By c k ( X ) (resp. c B k ( X )) we shall denote the sma llest size of a subset C ⊂ X , cen tral for (Bor e l) k - partitions of X . If no such a set C exists, then we put c k ( X ) = ∞ (res p. c B k ( X ) = ∞ ) where ∞ is assumed to be greater than an y cardina l n um b er. It follo ws from the definition that c B k ( X ) ≤ c k ( X ). W e hav e a lot of information on the num b ers c B k ( E n ) a nd c k ( E n ) fo r E uclidean spaces E n , see [B 2 ]. In pa rticular, w e known that (1) c 2 ( E n ) = c B 2 ( E n ) = 3 for all n ≥ 2; (2) c 3 ( E 3 ) = c B 3 ( E 3 ) = 6; (3) 12 ≤ c B 4 ( E 4 ) ≤ c 4 ( E 4 ) ≤ 14; (4) n ( n + 1) / 2 ≤ c B n ( E n ) ≤ c n ( E n ) ≤ 2 n − 2 for ev ery n ≥ 3. Much less is known on the num b ers c B k ( H n ) and c B k ( H n ) in the hyperb olic case. Theorem 2 yields the upp er b ound c 2 ( H 2 ) ≤ 3. In fac t, 3 is the exact v alue of c 2 ( H n ) fo r a ll n ≥ 2. Prop ositi on 1. c B 2 ( H n ) = c 2 ( H n ) = 3 for al l n ≥ 2 . Pr o of. The upper b ound c 2 ( H n ) ≤ c 2 ( H 2 ) ≤ 3 follows from Theorem 2. The low er b ound 3 ≤ c B 2 ( H n ) will follow as so on a s for an y tw o points c 1 , c 2 ∈ H n we construct a partition H n = A 1 ∪ A 2 in tw o Bo rel pieces containing no unbounded set, symmetr ic with resp ect to a p oint c i . T o co nstruct such a pa rtition, consider the line l co ntaining the p oints c 1 , c 2 and decomp ose l into t wo half-lines l = l 1 ⊔ l 2 . Next, let H b e a n ( n − 1)-hyper plane in H n , or thogonal to the line l . Let S b e the unit s phere in H centered at the intersection p oint of l and H . Let S = B 1 ∪ B 2 6 T ARAS BANAKH, AR TEM DUDKO, AND DU ˇ SAN REP OV ˇ S be a partitio n of S into t wo Borel pieces such that no an tipo dal p oints of S lie in the same cell of the partition. F or ea ch p oint x ∈ H n \ l cons ider the h yp erb olic plane P x containing the p oints x, c 1 , c 2 . The complement P x \ l deco mp o ses into t wo half-planes P + x ∪ P − x where P + x is the half-plane containing the p oint x . The plane P x int ersects the hyperpla ne H by a hyper bo lic line con taining tw o p oints of the s phere S . Fina lly put A i = l i ∪ { x ∈ H 2 \ l : P + x ∩ B i 6 = ∅} for i ∈ { 1 , 2 } . It is easy to chec k that A 1 ⊔ A 2 = H n is the desired pa rtition o f the hyperb olic space int o tw o Borel pieces none of which co ntains a n un bo unded subset symmetric w ith resp ect to one of the po int s c 1 , c 2 .  The preceding pr op osition implies that the cardinal num b er s c 2 ( H n ) a re finite. Problem 1. F or which numb ers k , n ar e the c ar dinal numb ers c k ( H n ) and c B k ( H n ) finite? Is it true for al l k ≤ n ? Except for the equality c 2 ( E n ) = 3, we hav e no informatio n on the num b er s c k ( E n ) with k < n . Problem 2. Calculate (or at le ast evaluate) the numb ers c k ( E n ) and c k ( H n ) for 2 < k < n . In all the cases where we know the exact v alues of the num b e rs c k ( E n ) and c B k ( E n ) we see that those n um b ers are equal. Problem 3. A r e the num b ers c k ( E n ) and c B k ( E n ) (r esp. c k ( H n ) and c B k ( H n ) ) e qual for al l k , n ? Having in mind that each subset not lying on a line is central for 2- partitions of the E uclide a n plane, we may ask ab out the sa me prop erty of the Lobachevsky plane. Problem 4. Is any subset C ⊂ H 2 not lying on a line c entr al for (Bor el) 2- p artitions of the L ob achevsky plane H 2 ? Finally , let us ask ab out the num b er s c B k ( H 2 ) and c k ( H 2 ). O bserve that The- orem 1 g uarantees that c B k ( H 2 ) ≤ c for all k ∈ N . Insp ecting the pro o f we c a n see that this upp e r b ound can be improved to c B k ( H 2 ) ≤ no n( M ) where non( M ) is the smallest cardinality of a non-meag er subset of the real line. It is cle a r that ℵ 1 ≤ non( M ) ≤ c . The exact lo c a tion of the car dinal non( M ) on the interv al [ ℵ 1 , c ] dep ends on axioms of Set Theor y , see [Bl]. In particular , the inequality ℵ 1 = no n( M ) < c is consistent with ZFC. Problem 5. Is the ine quality c B k ( H 2 ) ≤ ℵ 1 pr ovable in ZFC? Ar e the c ar dinals c B k ( H 2 ) c oun table? fin ite? The latter problem asks if H 2 contains a co untable (or finite) c e ntral set for Borel k -pa rtitions of the Lobachevsky pla ne. Insp ecting the pr o of of Theo rem 1 we can se e that it gives an “appr oximate” answer to this pr o blem: Prop ositi on 2. F or any k ∈ N ther e is a finite su bset C ⊂ H 2 of c ar dinality | C | ≤ k ( k + 1) / 2 such that for any p artition H 2 = B 1 ∪ · · · ∪ B k of H 2 into k Bor el pie c es and for any op en n eighb orho o d O ( C ) ⊂ H 2 of C one of the pie c es B i c ontains an unb oun de d subset S ⊂ B i symmetric with re sp e ct to some p oint c ∈ O ( C ) . ON COLORINGS OF THE LOBACHEVSKY PLANE 7 Remark 1. F or further results and op en problems rela ted to s ymmetry and color - ings s ee the surveys [BP 2 ], [BVV] and the lis t o f pro blems [BBGRZ, § 4]. References [B 1 ] T.Banakh, Solutions of c ertain pr oblems of I.V.Pr otasov fr om the c ombinatorics of c olorings , U Sviti Matemat yky , 3 :1 (1997), 8–11 (in Ukrainian). [B 2 ] T.Banakh, On a ca r dinal gr oup invariant r elate d t o p artition of ab elian gr oups , Mat. Za- metki. 64:3 (1998) , 341–350. [BBGRZ] T.Banakh, B. Bok alo, I.Guran, T.Radul, M.Zar ich nyi, Pr oblems fr om the L viv top olo g- ic al seminar , in: Op en Problems in T opology , I I (E.Pearl ed.), Elsevier, 2007, P . 655–667. [BP 1 ] T.Banakh, I.Protasov , Asymmetric p artitions of ab elian g r oups , M at. Zametki, 66 (1999), 10–19. [BP 2 ] T.Banakh, I.Protasov , Symmetry and c olorings: som e r esults and op en pr oblems , Izv. Gomel Univ. V oprosy Algebry . 4 (2001), 5–16. [BVV] T.Banakh, O.V erbitsky , Y a.V or obets, A R amsey tr e atment of sy mmetry , 2000, Electronic J. of Combinat orics. 7:1 (2000) R52. – 25p. [Bl] A.Bl ass , Combinatorial Car dinal Char acterist ics of the Continuum , in: Handb o ok of Set Theory (M. F oreman, M . Magidor, A. Kanamoried, eds.), Springer, 2007. [Ke] A.Kechris, Classical Descriptive Set Theory , Springer, 1995. (T. Banakh) Dep ar tment of Ma thema tics, L viv Na tional University, L viv, Ukraine, andInstytut Matema tyki, Akademia ´ Swie ¸ tokrzyska, Kielce, Poland E-mail addr e ss : tbanakh@yahoo .com (A. Dudko) Kharkiv Na tiona l University, Kharkiv, Ukraine E-mail addr e ss : artemdudko@ra mbler.ru (D. Rep o v ˇ s) Institute for Ma thema tics, Physics an d Mechanics, University of Ljubl- jana, Jadranska 19, Ljubljana, S lovenia E-mail addr e ss : dusan.repovs@ fmf.uni-lj.si