Coloring Questions on Axis-Parallel Rectangles and Arithmetic Progressions

Coloring Questions on Axis-P arallel Rectangles and Arithmetic Progressions Gáb or Damásdi F ebruary 23, 2026 Abstract W e presen t an explicit family of hypergraphs with arbitrarily large uniformity and c hromatic n umber that admit realizations in both geometric and n umber-theoretic settings. As an applica- tion, w e giv e a new proof of a theorem of Chen, Pac h, Szegedy , and T ardos. They show ed that for an y constan ts c, k ≥ 1 , there exists a finite p oint set P in the plane with the following property: for ev ery coloring of P with c colors, there is an axis-parallel rectangle containing at least k p oin ts, all of the same color. Their original pro of is probabilistic; we present an explicit construction. More- o ver, in the case k = 2 , w e show that one can even realize a graph that has arbitrarily large girth and c hromatic num b er sim ultaneously . W e also answer a question of P álvölgyi on coloring sets of in tegers with resp ect to certain finite arithmetic progressions. Finally , we giv e an application to coloring partially ordered sets. 1 In tro duction Coloring questions on graphs and h yp ergraphs arising from geometric or num b er-theoretic configura- tions play an imp ortant role in combinatorics. In this pap er we presen t a family of h yp ergraphs of large chromatic num b er that app ear naturally in b oth scenarios. A proper c -coloring of a hypergraph H = ( V , E ) is a map V → [ c ] such that no edge is mono chro- matic. The chr omatic numb er of H is the smallest c for whic h suc h a coloring exists. Giv en a set P of p oin ts in the plane and a family F of planar sets, let H ( P , F ) denote the hypergraph giv en b y incidence, i.e., the vertex set is P and the edge set is { P ∩ F | F ∈ F } . If the members of F b elong to a geometric family (e.g., disks, rectangles, conv ex sets), then w e say that H ( P , F ) is r e alize d b y that family . Coloring questions on geometric hypergraphs ha ve a v ast literature, see [ 5 ] for a recent surv ey and the webpage [ 13 ] for up-to-date results. Some of the families that will b e of interest to us are: • Axis-parallel rectangles: sets of the form { ( x, y ) | x 1 ≤ x ≤ x 2 , y 1 ≤ y ≤ y 2 } for some x 1 , x 2 , y 1 , y 2 ∈ R . • Bottomless axis-parallel rectangles: sets of the form { ( x, y ) | x 1 ≤ x ≤ x 2 , y ≤ y 2 } for some x 1 , x 2 , y 2 ∈ R . • Horizon tal strips: sets of the form { ( x, y ) | y 1 ≤ y ≤ y 2 } for some y 1 , y 2 ∈ R . In this pap er, we fo cus on the case where F consists of axis-parallel rectangles. This was first considered by Chen, Pac h, Szegedy , and T ardos [ 4 ]. Using a probabilistic argumen t, they show ed that w e can realize h yp ergraphs of arbitrarily large chromatic num ber and uniformity . Theorem 1.1 (Chen, Pac h, Szegedy , and T ardos [ 4 ]) . F or any c onstants c, k ≥ 1 , ther e exists a finite set P of p oints in the plane with the fol lowing pr op erty: for every c oloring of P with c c olors, ther e is an axis-p ar al lel r e ctangle c ontaining k p oints, al l of the same c olor. F or an y k ≥ 1 , Chek an and Ueck erdt ga ve an explicit set P of p oints and a family F of horizon tal strips and bottomless axis-parallel rectangles such that H ( P, F ) is k -uniform and not 2-colorable [ 3 ]. Since strips and b ottomless rectangles can b e replaced by ordinary rectangles without changing the incidence structure, their construction yields an explicit solution to Theorem 1.1 in the case c = 2 . W e generalize their construction to all c > 2 and pro ve a slightly stronger statement. A set of p oints 1 is called asc ending if the x -coordinate order of the points is the same as the y -co ordinate order of the p oin ts. W e sa y that a family of interv als I is neste d if for any A, B ∈ I we hav e A ⊆ B , B ⊆ A or A ∩ B = ∅ . Theorem 1.2. F or any c onstants c, k ≥ 1 , ther e exists an explicit finite set P of p oints and a family R of axis-p ar al lel r e ctangles in the plane with the fol lowing pr op erty: the y -pr oje ctions of the r e ctangles in R form a neste d family, and for every c oloring of P with c c olors, ther e is a r e ctangle in R that c ontains k asc ending p oints of P , al l of the same c olor. Finding graphs and hypergraphs that hav e large girth and chromatic n umber is a classical topic, going back to T utte [ 9 ] (under the pseudonym Blanche Descartes) and Erdős [ 10 ]. Large-girth con- structions were recently found in some geometric problems; see, for example, [ 7 , 8 ]. Using a graph construction from [ 14 ] we show a large-girth v ariant of Theorem 1.1 in the graph case k = 2 . Theorem 1.3. F or any c onstants c, g ≥ 1 , ther e exists a finite set P of p oints and a set R of axis- p ar al lel r e ctangles in the plane such that H ( P , R ) is a gr aph, has girth at le ast g and chr omatic numb er at le ast c + 1 . 1.1 Arithmetic progressions and a question of Pálv ölgyi One of the most famous coloring results in combinatorial n umber theory is v an der W aerden’s theorem [ 22 ]. It sa ys that for an y constants c, k ≥ 1 there is an n such that if w e color the in tegers from 1 to n using c colors, then w e alw ays find a mono chromatic arithmetic progression of length k . P álvölgyi prop osed the follo wing v arian t (see Problem 9.11 in [ 5 ]). F or D ⊂ N , let A D denote the family of all finite arithmetic progressions whose difference is in D . Instead of all arithmetic progressions, we will only consider arithmetic progressions from A D . On the other hand, we allow for restrictions on the base set. F ormally , for a base set V ⊂ Z and a family A of finite arithmetic progressions, let H ( V , A ) denote the incidence h yp ergraph on v ertex set V whose edge set is { A ∩ V | A ∈ A} . If the mem b ers of A b elong to A D , then we say that H ( V , A ) is r e alize d by arithmetic progressions whose difference is in D . This setting is clearly analogous to the problem of axis-parallel rectangles. Instead of a planar set of p oin ts, w e hav e a set of integers. Instead of axis-parallel rectangles, we hav e arithmetic progressions (by pure coincidence, b oth can be abbreviated as AP). Again we are interested in the coloring prop erties of these h yp ergraphs. Problem 1.4 (P álvölgyi, Problem 9.11 in [ 5 ]) . F or which D ⊂ N and k ≥ 1 c an we r e alize k -uniform hyp er gr aphs of lar ge chr omatic numb er using arithmetic pr o gr essions whose differ enc e is in D ? P álvölgyi noted that for finite D the chromatic n umber is b ounded and depends only on the divisibilit y lattice of D . On the other hand, if D is infinite, almost nothing is known; see [ 2 ] for partial results. P álvölgyi also p oin ted out that the case D = { 2 i | i ∈ N } sho ws a connection to geometric h yp ergraphs, as we can realize all finite hypergraphs that are realizable by p oin ts and b ottomless rectangles in the plane (see the discussion at [ 20 ]). W e sho w that the connection is m uch stronger. In fact, this case is equiv alent to realizing h yp ergraphs using axis-parallel rectangles whose y -pro jection is nested. Theorem 1.5. A hyp er gr aph is r e alizable by a family of axis-p ar al lel r e ctangles whose y -pr oje ctions form a neste d family if and only if it is r e alizable by arithmetic pr o gr essions whose differ enc e is a p ower of 2. The connection relies on a mapping that uses the v an der Corput sequence. W e also extend one direction to arbitrary infinite difference sets. Theorem 1.6. Supp ose D ⊂ N is an infinite set. If a hyp er gr aph is r e alizable by a family of axis- p ar al lel r e ctangles whose y -pr oje ction is neste d, then it is r e alizable by arithmetic pr o gr essions whose differ enc e is in D . Theorem 1.2 and Theorem 1.6 immediately imply the following resolution of P álvölgyi’s problem. Theorem 1.7. F or any infinite D and any c onstants c, k ≥ 1 ther e exists a set V ⊂ Z such that for any c -c oloring of V ther e is a finite arithmetic pr o gr ession A whose differ enc e is in D such that A ∩ V is mono chr omatic and has size at le ast k . 2 1.2 Coloring Hasse diagrams of p osets F or a partially ordered set ( P , < ) its Hasse diagram is the graph on vertex set P where u < v is an edge if there is no w ∈ P such that u < w < v . As Hasse diagrams are triangle-free, it is interesting to consider their coloring prop erties. Erdős and Ha jnal constructed p osets whose Hasse diagrams ha ve large chromatic num b er [ 11 ]. Recently Suk and T omon [ 21 ] constructed Hasse diagrams with n vertices and chromatic num b er Ω( n 1 / 4 ) . Kříž and Nešetřil sho wed that there are 2-dimensional p osets such that their Hasse diagram has large chromatic num b er [ 15 ]. A standard wa y to pro duce p osets from a set of planar p oints is to consider the ordering p < q ⇔ x ( p ) < x ( q ) and y ( p ) < y ( q ) . This giv es a 2-dimensional p oset, and ev ery 2 -dimensional p oset can b e represented this w ay . Note that { p, q } is an edge of the Hasse diagram of ( P, < ) if and only if they form an ascending set and there is an axis-parallel rectangle R suc h that P ∩ R = { p, q } . Therefore, Theorem 1.2 implies the following strengthening of the result of Kříž and Nešetřil. Corollary 1.8. F or any c onstants k , c ≥ 1 , ther e exists a 2-dimensional p oset P with the fol lowing pr op erty: for every c oloring of P with c c olors, ther e is a mono chr omatic incr e asing p ath in the Hasse diagr am of P of length k . In Section 2 we present a construction of a family of large-chromatic hypergraphs. In Section 3 we sho w how to realize these using rectangles (Theorem 1.2 ). Finally , in Section 4 we show ho w to reduce P álvölgyi’s question to the problem on axis-parallel rectangles (Theorems 1.5 and 1.6 ). 2 Constructing the h yp ergraph In this section we define, for each c, k ≥ 1 a k -uniform hypergraph H c k suc h that every c -coloring of H c k yields a mono chromatic edge. As a w arm-up, we recall a simpler construction, the k -ary tree h yp ergraph. Supp ose T = ( V , E ) is a ro oted tree. Let H T denote the following hypergraph on vertex set V . F or eac h leaf, the v ertices of the unique ro ot-leaf path form an edge. W e will call these the p ath e dges . F or eac h non-leaf vertex v , the children of v form an edge, these are the sibling e dges . W e say that a ro oted tree is of depth k if eac h ro ot-leaf path contains exactly k vertices. If T is the tree of depth k where eac h non-leaf v ertex has k c hildren, then H T is a k -uniform h yp ergraph, known as the k -ary tr e e hyp er gr aph . It is easy to see that H T is not 2-colorable for an y T . Indeed, suppose that we ha v e a 2-coloring of the vertices. W e can either follow the ro ot’s color down a path edge, or w e get stuck at a vertex whose c hildren form a monochromatic sibling edge. The k -ary tree h yp ergraphs play ed an imp ortan t role in a num b er of problems on geometric h y- p ergraphs, see [ 1 , 6 , 19 ] and Section 2.2 in [ 5 ]. Unfortunately , if k is large enough, then they cannot b e realized by axis-parallel rectangles [ 16 ]. The path edges are realizable on their o wn, but w e cannot add the sibling edges. T o remedy this problem, we mo dify the construction. The idea is to take man y trees instead of just one. W e realize the path edges in eac h tree and w e will replace the sibling edges with a new type of edge called tr ansversal e dges . Each transv ersal edge takes at most one vertex from eac h tree, and using them we will sho w that w e can follow the color of the root down a path edge in at least one of our trees. The precise construction is as follows. F or each c, k ≥ 1 , we inductively construct a vertex-ordered k -uniform h yp ergraph H c k of chromatic n umber at least c + 1 . As mentioned, the construction is a generalization of a construction by Chek an and Uec kerdt for 2-colors [ 3 ], and w e mostly follo w their terminology . F or c = 1 w e tak e k vertices in arbitrary order and a single edge containing all of them. Let k ≥ 1 and c > 1 be fixed. By induction, H c − 1 k exists. Let m denote the n um b er of its vertices. W e start by building an auxiliary rooted forest consisting of m k ro oted trees. The v ertex set of H c k will b e partitioned into stages , eac h stage con tains at most one vertex from eac h tree. The v ertices of a giv en stage will b e at a fixed distance j from the ro ots of their corresponding tree, that we will call the level of the stage. Eac h stage S of level j has exactly m k − j v ertices and comes with a fixed linear ordering < S on these v ertices. F urthermore, w e regard the v ertices of a stage as partitioned into m k − j − 1 disjoin t blo cks of m consecutiv e v ertices in the ordering < S . W e will consider subsets of a stage that con tain exactly one vertex from each blo c k, hence the following definition will b e useful. F or an ordered set A = 3  Stages of level 1. Stage of level 0. Figure 1: The h yp ergraph H 2 2 and its realization by axis-parallel rectangles. { a 1 , a 2 , . . . , a tm } let f m ( A ) denote the family of subsets of A that con tain exactly one elemen t from eac h block { a im +1 , a im +2 , . . . , a im + m } , i = 0 , . . . , t − 1 . W e start building the forest by taking m k ordered v ertices, they form the unique stage of level 0. They also serve as the ro ots of the m k trees. Supp ose that we hav e already defined a stage S of level j < k − 1 . Then for each subset S ′ ∈ f m ( S ) we define a new stage T ( S ′ ) of level j + 1 on m k − j − 1 new v ertices. Each v ertex of S ′ receiv es exactly one child in T ( S ′ ) and the v ertices of T ( S ′ ) inherit the ordering of their corresponding parents. F or stages of lev el k − 1 the process stops. Hence, w e obtain a forest of m k trees, whose v ertices are partitioned into stages of level 0 , 1 , . . . , k − 1 . Let V denote the vertex set of the forest; this will b e the v ertex set of H c k . F or a vertex v ∈ V , let root ( v ) denote the ro ot of the tree that contains v . Let path ( v ) denote the vertex set of the unique path from v to r oot ( v ) . No w we are ready to define the edges of the h yp ergraph H c k . There are tw o t yp es of edges. Let E P = { path ( v ) | v app ears in a stage of lev el k − 1 } , that is, for each vertex v that app ears in a stage of level k − 1 , add path ( v ) as an edge to H c k . As b efore, these are called p ath e dges . Secondly , for each stage S of level j do the following. Recall that the v ertices of S are partitioned in to m k − j − 1 disjoin t blocks of m consecutiv e elements in the ordering < S . In each of these blo cks, add edges to form a copy of the vertex-ordered hypergraph H c − 1 k suc h that the v ertex ordering of H c − 1 k matc hes the restriction of < S to that blo ck. That is, w e realize m k − j − 1 disjoin t copies of H c − 1 k in the stage. The edges app earing in these will b e called tr ansversal h yp eredges, and the set of all transversal h yp eredges appearing in any stage is denoted b y E T . Finally we set H c k = ( V , E P ∪ E T ) , see Figure 1 for an example. Lemma 2.1. The k -uniform hyp er gr aph H c k = ( V , E P ∪ E T ) is not pr op erly c -c olor able. Pr o of. W e use induction on c . F or c = 1 , we ha ve a single edge; hence the hypergraph is not 1-colorable. F or c > 1 suppose that there is a proper c -coloring and fix a color, sa y red. W e claim that each stage S of lev el j contains at least m k − j − 1 red vertices, one from each blo c k of m p oints. In other words, there is a red set in f m ( S ) . Indeed, the transversal h yp eredges form m k − j − 1 disjoin t copies of H c − 1 k . By induction, eac h color must app ear in each of these copies, hence we find a red v ertex in eac h. H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k H c − 1 k S 0 S 1 S 2 Figure 2: Finding a mono c hromatic path edge in the k = 3 case. Grey rectangles indicate a cop y of H c − 1 k , red v ertices indicate the elemen ts in B j in each stage. Using this observ ation and induction on j , we show that for each j ∈ { 0 , . . . , k − 1 } there is a stage 4 S j of lev el j and a subset B j ∈ f m ( S j ) suc h that for each v ∈ B j the en tire path path ( v ) is red. See Figure 2 for an example. F or j = 0 , let S 0 b e the unique stage at lev el 0. It consists of m k − 1 blo c ks, eac h forming a copy of H c − 1 k , we pick a red vertex from each to obtain B 0 . Assume that we ha ve found B j for some j < k − 1 . As B j ∈ f m ( S j ) there is a child stage S j +1 = T ( B j ) . By the induction h yp othesis, we can pic k a red vertex in eac h blo ck of S j +1 , giving us B j +1 . Each v ertex of B j +1 has its parent in B j , so path ( v ) is red for all v ∈ B j +1 . Finally , we find B k − 1 inside a stage of level k − 1 . It contains m k − ( k − 1) − 1 = m 0 v ertices, i.e., a single v ertex v . Then path ( v ) is mono chromatic (red), a contradiction. Hence, there is no prop er c -coloring of the hypergraph H c k . 3 Geometric realization W e prov e Theorem 1.2 in tw o steps. First, w e show that there is a realization of H c k b y axis-parallel rectangles that con tain ascending sets. Then w e sho w ho w to achiev e nested y -pro jection. F or a p oint p ∈ R 2 , write x ( p ) , y ( p ) for its co ordinates. A set P = { p 1 , . . . , p n } ordered b y x - co ordinate is asc ending if y ( p 1 ) < · · · < y ( p n ) and desc ending if y ( p 1 ) > y ( p 2 ) > · · · > y ( p n ) . W e sa y that p is ab ove q if y ( p ) > y ( q ) and use analogous definitions for b elow, to the right, and to the left . Lemma 3.1. F or every c, k ≥ 1 , the hyp er gr aph H c k c an b e r e alize d by axis-p ar al lel r e ctangles such that e ach r e ctangle c ontains an asc ending set of k p oints. Pr o of. W e use induction on c . F or c = 1 the h yp ergraph H c k has a single edge, and hence w e can pic k an y set of k ascending p oints. F or c > 1 , we find the realization of H c k in tw o phases. In the first phase, we embed the vertices such that the following prop erties hold. a) The vertices of a given stage S lie on a horizontal line, the stage line of S , and are ordered according to < S along the line. b) The path edges are realizable b y axis-parallel rectangles in this embedding. Then, in the second phase, w e perturb the vertices in each stage so that the transv ersal h yp eredges also b ecome realizable. Phase 1 . W e start by placing the ro ots (the level-0 stage) on a horizontal line in order. Then iterativ ely pic k a stage S that is already embedded but whose child stages T 1 , T 2 , . . . , T r are not, and em b ed the child stages sim ultaneously in the follo wing w a y . Pick r distinct horizon tal lines b elo w the line of S and abov e the next stage line, if there is one. F or eac h i ∈ { 1 , . . . , r } place the vertices of T i on the i -th line so that each v ertex has the same x -co ordinate as its parent in S . (This also ensures that T i is ordered according to < T i along the stage line.) Next, for each v ertex v ∈ S slightly shift the children of v to the left in the following w a y . Let w b e the v ertex preceding v in the x -co ordinate ordering of the whole set of points, if there is any . W e shift the children of v so that each stays on its stage line, they form a descending set, and they stay b et ween v and w in the x -co ordinate ordering, see Figure 3 . Lines for new stages. Line of stage S v v  After shifting: w Next stage line Figure 3: The c hildren of v are placed b elo w v then shifted to the left to form a descending set. After this pro cess, each non-ro ot vertex is below and to the left of its parent, and hence each path ( v ) is an ascending set. F urthermore, when w e shift the v ertices to the left, the order of the v ertices within a stage do es not c hange. 5 F or eac h v ertex v , let R v b e the rectangle whose bottom-left corner is v and whose top-right corner is root ( v ) . The rectangle R v con tains path ( v ) , since the path is ascending. W e claim that it con tains no other v ertex. W e only need to show that any v ertex w / ∈ path ( v ) do es not lie in R v . As v lies betw een r oot ( v ) and the preceding root v ertex, w e ma y assu me that root ( v ) = root ( w ) . Let p denote the highest level v ertex in path ( v ) ∩ path ( w ) . Let v ′ and w ′ denote the tw o children of p in path ( v ) and path ( w ) , resp ectiv ely . By the definition of p , we kno w that v ′ and w ′ are distinct v ertices and that they were created when we pro cessed the stage con taining p . By construction, the c hildren of p are descending, hence v ′ is either ab o ve and to the left of w ′ or b elow and to the righ t of w ′ , see Figure 4 . p v 0 w 0 v w p w 0 v 0 w v Figure 4: The tw o cases dep ending on how v ′ and w ′ w ere placed. The gray regions indicate the p ossible p ositions of v ′ and w ′ in the plane. Supp ose v ′ is abov e and to the left of w ′ . All descendants of w ′ , including w , are b elow w ′ . On the other hand, all descendants of v ′ are placed later than w ′ , hence they are on lines ab o ve the line of w ′ . Therefore, v is ab ov e w ′ , and w cannot be in R v . Supp ose v ′ is b elow and to the righ t of w ′ . All descendants of w ′ , including w , are to the left of w ′ . As the descendants of v ′ are placed later than w ′ , they are all to the right of it. Hence v is to the righ t of w and w cannot b e in R v . Thus R v con tains exactly the path v ertices, and every path edge is realized. Phase 2. Next, we realize the transversal edges. Right now the vertices of a given stage S of lev el j lie on a horizontal line and their x -co ordinate order matches < S , that is, the m k − j − 1 copies of H c − 1 k that are formed by the transv ersal edges o ccupy disjoint consecutiv e blocks of m vertices. Note that if we hav e a realization of any hypergraph b y axis-parallel rectangles, then one can obtain new realizations by changing the p osition of the points. As long as the x and y -co ordinate orders of the p oints do not change, we can realize the same hypergraph using rectangles. F or each block, replace the m p oints by a v ery thin realization of H c − 1 k whose v ertices are close to the corresponding vertices in the block. If the copy is thin enough, the rectangles cannot con tain p oin ts from any other stage. If the p erturbation is small enough, the existing path edges do not change. Hence, w e hav e realized H c k b y axis-parallel rectangles. Each realizing rectangle contains an ascending k -set by construction in Phase 1 and b y induction in Phase 2. Next, w e show that there is a realization where the y -pro jections of the rectangles form a nested family . Recall that a family of interv als I is neste d if for any A, B ∈ I we hav e A ⊆ B , B ⊆ A or A ∩ B = ∅ . (W e allo w the family to contain an interv al more than once.) Lemma 3.2. Ther e is a r e alization of H c k with axis-p ar al lel r e ctangles such that the pr oje ction of the r e ctangles to the y -axis is neste d. Pr o of. W e mo dify the construction in Lemma 3.1 . Again w e pro ceed b y induction on c , the c = 1 case is trivial. When realizing path edges via rectangles R v , the top side of each rectangle lies on the lev el-0 stage line. Hence, all path rectangles share the same upp er y -endpoint in pro jection, hence their y -pro jections form a nested family . Enlarge eac h sligh tly so that no p oin t lies on an y rectangle b oundary . F or transversal edges, each is realized inside an arbitrarily thin horizon tal neighborho o d of a stage line. If this neighborho o d is thin enough, then the y -pro jection of any transv ersal rectangle is either fully con tained in the y -pro jection of a giv en path rectangle or disjoint from it (b ecause points are not on b oundaries). Pro jections from different stages are disjoin t b y construction. 6 Finally , within a single stage, w e may place the disjoin t copies of H c − 1 k at sligh tly different vertical offsets (still within the stage neighborho o d), so that the pro jection interv als of rectangles from different copies are disjoint. Inside eac h copy , nestedness holds by induction. Therefore, the y -pro jections form a nested family . 3.1 Large girth v arian t F or g ≥ 2 , a g -cycle of a hypergraph is an alternating sequence, ( v 1 , E 1 , v 2 , E 2 , . . . , v g , E g ) , of distinct v ertices v 1 , . . . , v g and distinct edges E 1 , . . . , E g suc h that v g , v 1 ∈ E g , and v i , v i +1 ∈ E i , for i = 1 , . . . , g − 1 . The girth of a hypergraph is the length of the smallest cycle in the hypergraph. W e sho w that in the graph case k = 2 w e can ac hiev e arbitrarily large girth. Let us recall a construction from [ 14 ] using our terminology of stages. F or each c, g ≥ 1 we construct an ordered graph G c ( g ) of girth at least g and chromatic num b er at least c + 1 . W e use induction on c . F or c = 1 , the graph K 2 w orks. F or c > 1 , fix an auxiliary hypergraph H = ( V H , E H ) of uniformit y | G c − 1 ( g ) | , girth g , and chromatic n umber at least c + 1 (Such h yp ergraphs exist by standard probabilistic argumen ts, see [ 12 ]). W e use H to define our stages. Create a stage of lev el 0 that con tains the v ertices of H , but, contrary to the previous construction, do not place any edges inside this stage. All of the remaining stages are going to b e of lev el 1. F or each edge S ∈ E H add a stage T ( S ) of level 1. It contains a child vertex v ′ for eac h v ∈ S . W e connect v and v ′ , and < T ( S ) is the ordering inherited from S . As H is | G c − 1 ( g ) | -uniform, T ( S ) contains | G c − 1 ( g ) | v ertices. Add edges in T ( S ) to realize a single copy of the graph G c − 1 ( g ) suc h that the ordering of the v ertices matc hes the ordering < T ( S ) . Lemma 3.3. The chr omatic numb er of G c ( g ) is at le ast c + 1 . Pr o of. W e pro ceed b y induction on c . F or c = 1 the c hromatic n um b er is 2. F or c ≥ 2 supp ose G c ( g ) admits a c -coloring. Since H has a c hromatic num b er of at least c + 1 , in any c -coloring of the stage of level 0 some edge S ∈ E H is mono chromatic. This implies that on stage T ( S ) one of the colors is forbidden. But T ( S ) spans a copy of G c − 1 ( g ) , wh ose chromatic n umber is at least c b y induction, a con tradiction. Lemma 3.4. The girth of G c ( g ) is at le ast g . Pr o of. F or any fixed g , we pro ceed by induction on c . F or c = 1 w e hav e chosen K 2 , a graph that con tains no cycles. F or c > 1 suppose that C = ( v 1 , . . . , v t ) is a cycle in G c ( g ) . If each v i is in the same stage of level 1, then t ≥ g by induction. Otherwise, let x 1 , x 2 , . . . , x m b e the vertices of C that lie in the level-0 stage, listed in cyclic order along C . There are no edges in the stage of level 0, nor b etw een the stages of level 1, so the part of C b et ween x i and x i +1 is non-empt y and must run in a single stage of lev el 1 corresp onding to some hyperedge E i ∈ E H . W e claim that there is a cycle of length at most m in H . Indeed, consider ( x 1 , E 1 , x 2 , E 2 , . . . , x m , E m ) . By construction x m , x 1 ∈ E m , and x i , x i +1 ∈ E i , for i = 1 , . . . , m − 1 . The x i v ertices are distinct. If the E i ’s are also distinct, then we hav e found a cycle of H . If the E i are not distinct, then fix a pair i, j where E i = E j , i  = j and | i − j | is minimal. Then ( x i +1 , E i +1 , x i +2 , E i +2 , . . . , x j , E j ) is a cycle of H of length at most m . Since H has girth at least g , we obtain t ≥ m ≥ g , that is, the length of C is at least g . The realization of G c ( g ) b y axis-parallel rectangles is essen tially the same as in Lemma 3.1 (place stages on separated horizon tal lines and realize each stage-internal cop y), we omit the details. This finishes the pro of of Theorem 1.3 . 4 Arithmetic progressions In this section w e turn our atten tion to coloring in tegers with resp ect to arithmetic progressions. After some preparation, w e show that realizabilit y by nested-pro jection rectangles is equiv alent to realizabilit y b y finite arithmetic progressions whose difference is a p ow er of 2. 7 W e will use the following notation for the closed horizon tal strip and the upw ard op en horizontal strip: S [ a, b ] = { ( x, y ) | a ≤ y ≤ b } S [ a, b ) = { ( x, y ) | a ≤ y < b } . The van der Corput sequence is a well-studied notion in discrepancy theory , see for example [ 17 ]. It is defined as follo ws. Supp ose the binary expansion of n ∈ N is n = ℓ X i =0 d i 2 i , d i ∈ { 0 , 1 } , then the n th term is a n = ℓ X i =0 d i 2 − i − 1 . W e will show that subsequences of the v an der Corput sequence pro vide an in terface betw een the t wo problems. The connection to coloring p oin ts with resp ect to rectangles is not new. See for example [ 18 ], and the following problem that w as prop osed b y Gáb or T ardos for the 2015 Sch w eitzer comp etition. Problem 4.1 (Miklós Sch weitzer Memorial Competition, 2015. Problem 2.) . L et { a n } b e the van der Corput se quenc e. L et P b e the set of p oints on the plane that have the form ( n, a n ) . L et G b e the gr aph with vertex set V that is c onne cting any two p oints ( p, q ) if ther e is an axis-p ar al lel r e ctangle R such that R ∩ V = { p, q } . Pr ove that the chr omatic numb er of G is finite. F or a nested family I , we can define a ro oted forest T I based on I in the following w ay . The v ertices of T I are the elements of I , and we connect A to B if A ⊆ B but there is no C ∈ I such that A ⊊ C ⊊ B . The ro ots are the maximal elemen ts. Equiv alen tly , T I is the Hasse diagram of the p oset ( I , ⊆ ) . W e say that a nested family is a p erfe ct neste d family of depth t if T I is a p erfect binary tree with t levels so that eac h non-leaf interv al is the union of its t wo children. Equiv alently , a nested family I is a p erfect nested family of depth t if we can index its elemen ts using the 0 / 1 -sequences of length at most t − 1 such that I s 1 s 2 ...s r = I s 1 s 2 ...s r 0 ∪ I s 1 s 2 ...s r 1 for all s 1 , . . . , s r ∈ { 0 , 1 } with r < t − 1 . F or example, for an y t ∈ N , the interv al family { [ i 2 k , i +1 2 k ) } ov er all 0 ≤ k < t and 0 ≤ i < 2 k is a p erfect nested family of depth t . Note that this is the same family as { [ a b , a b + 2 − k ) } ov er all 0 ≤ k < t and 0 ≤ b < 2 k . Observ ation 4.2. A ny neste d family c an b e extende d to a p erfe ct neste d family of depth t for some sufficiently lar ge t ∈ N . First, we fo cus on the D = { 2 i | i ∈ N } case. Let A 2 i denote all finite arithmetic progressions with difference from { 2 i | i ∈ N } . This case is essentially equiv alen t to realizing h yp ergraphs using nested axis-parallel rectangles. Pr o of of The or em 1.5 . (A rithmetic pr o gr essions ⇒ neste d r e ctangles) . First, supp ose V ⊂ Z . W e w ant to sho w that H ( V , A 2 i ) is realizable b y axis-parallel rectangles whose y -pro jections are nested. Consider the planar set of p oin ts P V = { ( n, a n ) | n ∈ V } , where a i denotes the i -th term of the v an der Corput sequence. W e claim that every edge of H ( V , A 2 i ) can be realized by an axis-parallel rectangle and the resulting family has nested y -pro jections. Consider the infinite arithmetic progression A = { 2 t k + b | k ∈ Z } for some fixed t ∈ N and b < 2 t . F or an y n ∈ N , w e ha ve a n = ℓ X i =0 d i 2 − i − 1 = t − 1 X i =0 d i 2 − i − 1 + ℓ X i = t d i 2 − i − 1 = a ( n mo d 2 t ) + ℓ X i = t d i 2 − i − 1 . Moreo ver, the v alues { a 0 , . . . , a 2 t − 1 } are spaced by at least 2 − t . Therefore, n ≡ b (mo d 2 t ) ⇐ ⇒ a b ≤ a n < a b + 2 − t . Th us the p oints { ( n, a n ) | n ∈ A ∩ V } are exactly the p oints of P V that lie in the strip S [ a b , a b + 2 − t ) . Cho osing a finite consecutiv e subset of A corresponds to in tersecting this strip with a vertical slab in x , i.e., to an axis-parallel rectangle. It is also easy to see that the family of interv als { [ a b , a b + 2 − t ) | 8 2k+1 2k+0 4k+0 2k+1 4k+2 4k+3 0 1 0001 0011 0111 1000 1010 1111 8k+7 8k+6 8k+5 8k+4 8k+3 8k+2 8k+1 8k+0 1 3 7 8 10 15 } S [ a 1 , a 1 + 1 / 2] Figure 5: The p oint set giv en by the v an der Corput sequence for V = { 1 , 3 , 7 , 8 , 10 , 15 } . The grey rectangle captures A ∩ V for the arithmetic progression A = 3 , 5 , 7 . b, t ∈ N and b < 2 t ] } is nested, see Figure 5 . Hence, w e can represent each edge using an upw ard open rectangle, and w e can choose them such that their y -pro jection is nested. (Neste d r e ctangles ⇒ arithmetic pr o gr essions ) . Let P = { p 1 , . . . , p n } ⊂ R 2 b e ordered by increasing x -co ordinate, and let R b e a family of axis-parallel rectangles whose y -pro jections form a nested family I . By Observ ation 4.2 , extend I to a p erfect nested family I ′ of depth t . Index I ′ b y binary strings of length at most t − 1 as { I s 1 ··· s r } , such that I s 1 s 2 ...s r = I s 1 s 2 ...s r 0 ∪ I s 1 s 2 ...s r 1 for all s 1 , . . . , s r ∈ { 0 , 1 } with r < t − 1 . F or eac h point p i = ( x i , y i ) , since I ′ partitions its top interv al into leaf in terv als, there is a unique leaf I s 1 ··· s t − 1 con taining y i . Define v i =  t − 1 X j =1 s j 2 j − 1  + i 2 t − 1 , V = { v 1 , . . . , v n } ⊂ N . Because the first term is < 2 t − 1 , we hav e v 1 < · · · < v n , so the order on V matc hes the x -order on P . No w consider a strip corresp onding to an in terv al I q 1 ··· q r (with r ≤ t − 1 ). Let A = n k 2 r + r X j =1 q j 2 j − 1    k ∈ Z o . This is an arithmetic progression with difference 2 r . Consider a p oin t p i = ( x i , y i ) and let I s 1 ...s t − 1 denote the leaf in terv al that con tains y i . Since I ′ is p erfect, p i is in the strip S [ I q 1 q 2 ...q r ] if and only if q 1 . . . q r is a prefix of s 1 . . . s t − 1 . That is, if and only if v i is of the form v i =  r X j =1 q j 2 j − 1  +  t − 1 X j = r +1 s j 2 j − 1  + i 2 t − 1 =  r X j =1 q j 2 j − 1  + 2 r  t − 1 X j = r +1 s j 2 j − 1 − r  + i 2 t − 1 − r  . Hence, this prefix condition is equiv alent to v i ≡ P r j =1 q j 2 j − 1 (mo d 2 r ) , i.e., to v i ∈ A . Hence in tersections of strips (and therefore rectangles) with P corresp ond to intersections of p ow er-of-tw o progressions with V . 4.1 The general case: Pro of of Theorem 1.6 Pr o of of The or em 1.6 . Let D ⊂ N be infinite. W e will use a rapidly gro wing subsequence of D . 9 Define d 1 < d 2 < · · · recursively as follo ws: d i = min n d ∈ D : d > 2 i − 1 · lcm( d 1 , . . . , d i − 1 ) o . Then for eac h i ≥ 1 we hav e lcm( d 1 , . . . , d i +1 ) ≥ d i +1 > 2 i lcm( d 1 , . . . , d i ) . (1) This ensures that certain mo dular equations are solv able. The follo wing lemma captures this prop ert y . Lemma 4.3. L et d 1 < d 2 < · · · satisfy ( 1 ) . If the mo dular system { x ≡ r i (mo d d i ) } j i =1 has a solution, then ther e ar e at le ast 2 j distinct choic es of r j +1 ∈ { 0 , 1 , . . . , d j +1 − 1 } such that { x ≡ r i (mo d d i ) } j +1 i =1 is solvable. Pr o of. Let L = lcm( d 1 , . . . , d j ) and supp ose the system has a solution, i.e., a residue class x ≡ r (mo d L ) . Consider ℓ i = r + iL, i = 0 , 1 , . . . , 2 j − 1 . Eac h ℓ i solv es the first j congruences. Cho ose r j +1 ≡ ℓ i (mo d d j +1 ) ; then ℓ i is a solution of the extended system. W e claim the residues ℓ i mo d d j +1 are all distinct for 0 ≤ i < 2 j . If ℓ i 1 ≡ ℓ i 2 (mo d d j +1 ) for i 1  = i 2 , then d j +1 | ( ℓ i 1 − ℓ i 2 ) . But L | ( ℓ i 1 − ℓ i 2 ) = ( i 1 − i 2 ) L as well, so lcm( L, d j +1 ) = lcm( d 1 , . . . , d j +1 ) divides the difference. On the other hand, 0 < | ℓ i 1 − ℓ i 2 | ≤ (2 j − 1) L < 2 j L < lcm( d 1 , . . . , d j +1 ) , where the strict inequality uses ( 1 ). This con tradiction shows distinctness, giving at least 2 j c hoices. Let P = { p 1 , . . . , p n } ⊂ R 2 b e ordered by increasing x -coordinate, and let R b e a family of axis- parallel rectangles whose y -pro jections form a nested family I . By Observ ation 4.2 , extend I to a p erfect nested family I ′ of depth t . Index I ′ b y binary strings of length at most t − 1 as { I s 1 ··· s r } , suc h that I s 1 s 2 ...s r = I s 1 s 2 ...s r 0 ∪ I s 1 s 2 ...s r 1 for all s 1 , . . . , s r ∈ { 0 , 1 } with r < t − 1 . F or eac h in terv al I s 1 ··· s m at depth m we will c ho ose a residue r s 1 ··· s m ∈ { 0 , 1 , . . . , d m − 1 } suc h that: (i) for every string s 1 · · · s m , the system { x ≡ r s 1 ··· s j (mo d d j ) } m j =1 is solv able; (ii) for each fixed m , the 2 m v alues { r s 1 ··· s m } are pairwise distinct. W e c ho ose these residues inductively o ver m . F or the step m → m + 1 , eac h solv able system at depth m has at least 2 m extensions b y Lemma 4.3 ; w e choose distinct residues for the children to satisfy (ii). F or each leaf string s 1 · · · s t − 1 , fix one integer f s 1 ··· s t − 1 < lcm( d 1 , . . . , d t − 1 ) solving { x ≡ r s 1 ··· s j (mo d d j ) } t − 1 j =1 . F or eac h p i = ( x i , y i ) , let I s 1 ··· s t − 1 b e the unique leaf in terv al con taining y i and set v i = f s 1 ··· s t − 1 + i · lcm( d 1 , . . . , d t − 1 ) , V = { v 1 , . . . , v n } ⊂ Z . Since f s 1 ··· s t − 1 < lcm( d 1 , . . . , d t − 1 ) , we ha ve v 1 < · · · < v n . Hence the order on V matches the x -order on P . Finally , consider any strip corresp onding to I q 1 ··· q r (with r ≤ t − 1 ). Let A b e the set of integers satisfying the congruence x ≡ r q 1 ··· q r (mo d d r ) . This set is an arithmetic progression with common difference d r ∈ D . Consider a p oint p i = ( x i , y i ) and let I s 1 ...s t − 1 denote the leaf interv al that contains y i . Since I ′ is perfect, p i is in the strip S [ I q 1 q 2 ...q r ] if and only if q 1 . . . q r is a prefix of s 1 . . . s t − 1 . The in teger v i is a solution of { x ≡ r s 1 ··· s j (mo d d j ) } t − 1 j =1 . Hence, if q 1 . . . q r is a prefix of s 1 . . . s t − 1 , then v i is a solution of x ≡ r q 1 ··· q r (mo d d r ) . On the other hand, if it is not a prefix, then v i is a solution of x ≡ r s 1 ··· s r (mo d d r ) , and by ( ii ) w e hav e r s 1 ··· s r  = r q 1 ··· q r . Hence, p i is in the strip S [ I q 1 q 2 ...q r ] if and only if v i is in the arithmetic progression A . Therefore strip (and hence rectangle) in tersections corresp ond to in tersections with arithmetic progressions of allo wed differences, completing the realization. 10 5 Final remarks As we hav e seen, for k = 2 w e hav e constructions of large girth. The main reason why the same ideas do not w ork for k = 3 is that certain pairs of path edges of a tree o verlap in more than one vertex. Problem 5.1. Is it true that for any c onstants c, k , g ≥ 1 ther e exists a finite set of p oints and a finite set of axis-p ar al lel r e ctangles such that their incidenc e hyp er gr aph is k -uniform, has chr omatic numb er at le ast c and girth at le ast g ? W e note that ev en the following easier question is open. Problem 5.2. Is it true that for any c ≥ 1 ther e exists a finite set of p oints and a finite set of axis- p ar al lel r e ctangles such that their incidenc e hyp er gr aph is 3 -uniform, has chr omatic numb er at le ast c and any two e dges interse ct in at most one vertex? A c kno wledgmen ts This pro ject was initiated at the Early Career Researchers in Combinatorics 2024 workshop in Edin- burgh. The author thanks the organizers for creating a stimulating research environmen t. 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