Octants are Cover Decomposable

Octan ts are Co v er Decomp osable Balázs Kesz egh and Dömötör P álv ölgyi Septem b er 27, 2018 Abstract W e pro ve that octan ts are co ver-dec omp osable, i.e., an y 12 -fold co vering of an y subset of the space with a finite num b er of translates of a giv en o ctant can b e decomp osed in to t wo co verings. As a corollary , w e obtain that any 12 -fold co v ering of any subset of the plane with a finite n um b er of homothetic copie s of a giv en triangle can be decomposed in to tw o co verings. W e also show that any 12 -fold cov ering of the whole plane with op en triangles can b e decompos ed int o t wo co verings. How ev er, w e ex hibit an indecomposable 3 -fold co vering. 1 In tro duction Let P = { P i | i ∈ I } b e a colle ction of geome tric sets in R d . W e sa y that P is an m -fold c overing of a set S if ev ery p oin t of S is con tained in at least m mem b ers of P . A 1 - fold co v ering is simply called a c ov e ring . Definition. A geometric set P ⊂ R d is said to be c over-de c omp osable ∗ if there exists a (m inimal) constan t m = m ( P ) such that ev ery m -fold cov ering of any subset of R d with a finite n um b er of trans lates of P can b e decompo sed in to t w o co v erings of the sam e subs et. Defin e m as the co ver-dec omp osabilit y constan t o f P . The simples t ob jects to examine are the orthan ts of R d . It is eas y to see that a quadran t ( 2 -dimensional orthan t) is cov er-decomp o sable. Cardinal [2] noticed that orthan ts in 4 and higher dimensions are not co v er-decomp osable as there is a plane on whic h their trace can b e any family of axis-arallel rectangles and it was sho wn b y P ac h, T ardos and Tóth [7] that suc h families migh t not b e decomp osable in t o tw o co v erings. Cardinal ask ed whether o ctan ts ( 3 -dimensional orthants) are cov er-decomposable. Our main result is an affirmativ e answ er (the pro of is in Section 2). ∗ In the literatur e the definition is slightly different and the notion defined here is sometimes called finite- c over-de c omp osable , how ever, to avoid unnecessa r y complications, w e simply use c over-de c omp osable . 1 Theorem 1. Octants ar e c over-de c omp osable, i.e., any 1 2 -fold c overing of any subset o f R 3 with a finite n umb er o f tr anslates of a g i v en o ctant c an b e de c omp ose d into two c ov erings. The in tersection of the translates of the o ctant con taining ( −∞ , −∞ , −∞ ) with the x + y + z = 0 plane giv es the homothetic copies of an equilateral triangle. Since any triangle can b e obtained b y an affine transformation of the equilateral triangle w e obtain Corollary 2. Any 12 -fold c overin g of any subset of the plane with a finite n umb er of ho mothetic c opies of any give n triangle c an b e de c o m p ose d in to two c ove rings . Using s tandard compactnes s argumen ts, this implies the fo llo wing (the pro of is in Section 4): Theorem 3. Any lo c al ly finite ∗ , 12 -fold c o v ering of the whole plane with homothetic c opies o f a triangle is de c omp osable into two c ove rin gs. The analo g s of Corollary 2 and Theorem 3 for translates o f a giv en triangle w ere pro v ed with a bigg er constan t by T ardos and Tóth [8] using a more complicated ar g umen t † . F ollowin g their idea, using Theorem 3 for tra nslates of a giv en o p en triangle we obtain Corollary 4. Any 12 -fold c overing of the whole pl a ne with the tr anslates of an o p en triangle is de c omp o sable into two c overin gs. Our result brings the task to determine t he exact cov er-decompo sabilit y constant of triangles in range. T ardos and Tóth state that they cannot ev en rule out the p ossibilit y that the co v er- decompo sability constan t of tria ng les is 3 . T o complemen t our upp er b ound, in Section 3 w e sho w a construction proving that the constant is actually at least 4 . Our pro o f of Theorem 1 in fact prov es the following equiv alen t, dual ‡ form of Theorem 1. Theorem 5. Any finite set of p oints in R 3 c an b e c olor e d with two c olors such that an y tr ansla te of a given o ctant with at le a st 12 p oi n ts c on tains b oth c olors. Finally , w e men tion the dual of Coro lla r y 2 , whic h is not equiv alen t to Corollary 2 but follo ws from Theorem 5 the same w ay as Corollary 2 follow s fr o m Theorem 1. Corollary 6. Any finite planar p o int set c an b e c olor e d with two c olors such that any hom othetic c opy of a gi v en triangle that c ontains at l e ast 12 p oints c o n tains b oth c olors. F or more results on cov er-decomposability , see the recen t surv eys [5] and [6]. ∗ W e sa y that a covering is lo c al ly finite if every compact set in tersects only a finite num b er o f co vering sets, i.e. ho mothetic copies of the giv en triangle, in our case . † The original pro of gav e m = 43 which was later improv ed by Á cs [1] to m = 19 which is still worse than our 1 2 . ‡ F or more on dua lization a nd for the pro o f of e quiv alence, se e the surveys [5] and [6]. 2 2 Pro of of Theore m 1 and 5 Denote b y W the o ctan t with a p ex at the origin con taining ( −∞ , −∞ , −∞ ) . W e will work in the dual setting, that is we ha ve a finite set of p oints, P , in the space, that we w ant to color with t w o colors suc h that an y translate of W with at least 12 points c on tains b oth colors. If w e can do this for any P , then it follow s using a standard dualization argument (see [5] or [6]) that W (and thu s an y o ctan t) is co ver-dec omp osable. So from no w on our goa l will b e to sho w the existence of suc h a coloring. F or simplicit y , supp ose that no num b er o ccurs m ultiple times among the co ordinates of the p oin ts of P (o therwise, by a small p erturbation of P w e can get suc h a p oin t set, a nd its coloring will b e also g o o d for P ). Denote the point of P with the t th smallest z co ordinate b y p t and the union of p 1 , . . . , p t b y P t . First w e w ill show ho w to reduce the coloring of P t to a planar and th us more trac ktable problem. Denote the pro jection of P on the z = 0 plane by P ′ . Similarly denote the pro jection of p t b y p ′ t , the pro jection of P t b y P ′ t and the pro jection of W by W ′ . Therefore W ′ is the quadran t with ap ex at the orig in con taining ( −∞ , −∞ ) . Claim 7. W e c an c olor the p oin ts of the pla nar p oin t s e t, P ′ , with two c olors such that for any t and a n y tr anslate of W ′ c ontaining at le ast 12 p oints of P ′ t , it is tr ue t hat the interse ction of this tr ansla te and P ′ t c ontains b oth c olors if and on l y if we c an c olor the p oints of the sp atial p oint set, P , with two c o lors such t hat for any t r anslate of W c ontaining at le ast 12 p o ints of P , it is true that the interse ction of this tr ansla te an d P c ontains b oth c ol o rs. Pr o of. Clearly , if w e take a tra nslate of W with ap ex w having z co ordinate bigger than the z co ordinate of p t and smaller t han the z co ordinate o f p t +1 , then the pro jection of the in tersection of this tra nslate with P is equal to the in tersection of P ′ t with the tra nslate of W ′ ha ving ap ex w ′ , the pro jection of w . Th us ha ving a go od coloring fo r one problem g iv es a go o d coloring fo r the other if w e g ive p t and p ′ t the same color for ev ery t . In the rest o f this pap er, suc h a coloring of a planar p oin t set is called a go o d c ol o ring . No w w e will pro ve that an y P ′ has a go o d coloring, thus establishing Theorem 5 and since they a re equiv alen t, also Theorem 1. T o a v oid going mad, w e will omit the ap ostrophe in the following, so w e will simply write W instead of W ′ and so o n. Also, w e will use the term we dge to denote a translate of W . A p o ssible w ay to imagine this planar problem is that in ev ery step t w e ha v e a set of p oin ts, P t , a nd our goal is to color the comi ng new p oint, p t +1 , suc h that w e alw a ys ha v e a go o d coloring. W e note that this w ould b e impo ssible in an onlin e setting, i.e. without k no wing in adv ance whic h p oin ts will come in whic h o r der. But using that w e kno w in adv ance ev ery p i mak es the problem solv able. W e start b y in tro ducing s ome notation. If p x < q x but p y > q y then w e sa y that p is NW from q and q is SE f rom p . In thi s case w e call p and q incomparable. Similarly , p is SW from 3 q (and q i s NE from p ) if and only if b oth co ordinates o f p are smaller than the resp ectiv e co ordinates of q . Instead of coloring the points, w e will rather define on them a bipartite gra ph G , whose prop er tw o-coloring will giv e us a go o d coloring. A ctually , as w e will la t er se e, this graph will b e a forest. W e define G recu rsiv ely , starting with the empt y set and the empt y graph. At an y step j w e define a graph G j on the points of P j and also main tain a set S j of pairwise incomparable p oin ts, called the stair c ase . Th us, b efore the t th step w e ha v e a gra ph G t − 1 on the p oints of P t − 1 and a set S t − 1 of pairwise incomparable p oints. In the t th step we add p t to our p oin t set obtaining P t and w e will define the new staircase, S t , and also the new graph, G t , whic h will hav e G t − 1 as a subgraph. Before the exact definition o f S t and G t , we mak e some more definitions and fix some prop ertie s that will b e main tained during the pro ces s. In an y step j , w e sa y that a p oin t p is go o d if any wedge containing p a lready con tains tw o p oin ts of P j connected by a n edge of G j . I.e. at an y time af ter j a w edge con taining p will con tain p oints of b ot h colors in the final coloring. A t an y time j , consider the order of the points of S j giv en by their x c o or dinates. If tw o p oin ts of S j are consec utiv e in this order then w e sa y that these staircase p oin ts are neighb ors ∗ . A p oint s of the staircase is almost g o o d if a ny w edge containin g s and its neigh b or(s) con tains t wo p oints of P j connected b y an edge of G j . Notice that the go od p oin ts and the neighbors of the go o d p oin ts are alwa ys almost go o d. W e say that a p oin t p o f P j is ab ove the staircase if there exists a staircase p oint s ∈ S j suc h that p is NE f r o m s . If p is not ab ov e or on the staircase, then we say that p is b elow the staircase. No w w e can state the prop erties w e maintain: A t an y time j : 1. All p o in ts ab ov e the staircase are go od. 2. All p o in ts of the staircase are almost g o o d. 3. All p o in ts b elo w the staircase a re incomparable. 4. If a w edge only con tains p oints that are below the staircase then it con tains at most 3 p oin ts. F or t = 0 all these prop erties are triv ially true. Supp ose that the prop erties hold at time t − 1 . No w w e proceed w ith po int p t according to the fo llo wing algorithm maintaini ng all the prop erties. During the pro cess, w e denote the actual graph b y G and the actual staircase b y S . A lgorithm Step t Set G = G t − 1 and S = S t − 1 . ∗ This does no t mea n that they are c o nnected in the graph. 4 q = p t p q = p t p p t Figure 1: Rep eated application of Step (b) of the algorithm Step (a) If p t is ab o v e the staircase S t − 1 then w e do the following, otherwise skip to Step (b). In this case S t = S t − 1 and G t = G t − 1 ∪ { e } , where e is an arbitrary edge b et w een p t and a p oin t s of S t − 1 whic h is SW from p t . The prop erties will hold trivially by induction, the only thing we need to che c k is if p t is a goo d p oin t, but this is again guaran teed as an y w edge con taining p t con tains t he edge e . The a lgorithm terminates. Note that w e pro ceed further if and only if p t is b elo w the staircase S t − 1 . Step (b) If there exist tw o p oints p and q that are b elo w the staircase and p and q are comparable then w e do the following, otherwise skip to Step (c). Without loss of generalit y supp ose that q is SW from p . Notice that b ecaus e of Prop ert y 3, either p or q is the last added p oin t and there are no p o in ts b elo w the stairc ase that are NE from p . No w, define the new staircase , S , as S min us the p oin ts of S that are NE fro m p , plus the p oin t p . This wa y the p oin ts o f the staircase r emain pairwise incomparable as w e a dded p and deleted all the p oin ts that were comparable t o p . Also, w e add the edge pq to the graph, i.e. G := G ∪ { pq } . F or an illustration of rep eated application of this step, see Figure 1 (edges o f G are drawn red). Th us, an y w edge con taining p con tains the edge pq , i.e. p is a go o d p oint. Prop ert y 1 is true for the points that we re abov e the old S b y induction. All other p oints a b ov e S are exactly the p oin ts that w ere deleted from the staircase in this step. Al l suc h p oin ts are NE fro m p and th us any w edge containing them contains the e dge pq . Property 2 holds for p as it is a go o d p oin t, it holds for the 2 neigh b ors of p as a n y p oin t neighboring a go od p o in t is an almost go o d p oin t. F or a ny other s from the staircase its neigh b o rs remain the same, so it remains almost go od. Go bac k to Step (b) until Prop erty 3 is satisfied, then pro ceed to Step (c). Step (c) If there exist 4 p oin ts b elow the staircase suc h that these 4 p oin ts are pairwise incompa- rable and there exists a w edge V suc h that V con tains these 4 p oin ts but no p oin ts of the staircase t hen do the following, otherwise skip to Step (d). Denote these 4 p oin ts b y q 1 , q 2 , q 3 , q 4 in increasing order of t heir x co or dinates. Notice that there are no p oin ts b elow the staircase that are comparable b ecause of Step (b). 5 q 3 q 1 q 2 q 4 q 3 q 1 q 2 q 4 (a) q 3 q 1 q 2 q 4 V (b) Figure 2: Application of Step (c) of the algorithm No w define the new S as the old S min us the p oin ts of S that are NE from q 2 or q 3 , plus the p oints q 2 and q 3 . This w a y the points of the staircase remain pairwise incomparable a s w e added q 2 and q 3 and deleted all the points that w ere comparable to them. Also, we add the edges q 1 q 2 and q 3 q 4 to the graph, i.e. G t = G t − 1 ∪ { q 1 q 2 , q 3 q 4 } . F or an illustration see Figure 2(a). Prop erty 1 is true f o r the p oints that w ere ab ov e the old S b y induction. All other p oin ts ab ov e the new S are exactly the p o in ts that w ere deleted from the staircase in this step. It is easy to c hec k that suc h a p oin t is either NE fro m b oth of q 1 and q 2 or it is NE from b oth of q 3 and q 4 (w e use that V w as completely below the staircase , see Figure 2(b)). Thu s, a w edge con ta ining suc h a p oin t con tains the edge q 1 q 2 or the edge q 3 q 4 , Prop erty 1 will b e true. Prop ert y 2 is true for q 2 and also for its neigh b or whic h is not q 3 as a w edge co v ering them m ust co v er q 1 as w ell and thus the edge q 1 q 2 , i.e. they are almost go o d. By symmetry q 3 and its neighbor whic h is not q 2 are also almost go od. F or an y other s from the rest of the staircase, s remains almost g o o d b y induction as its neigh b o rs do not change . Go bac k to Step (c) until Prop ert y 4 is satisfied, then pro ce ed to Step (d). Step (d) Set S t = S and G t = G a nd the algorithm terminates. A dding p t b elo w the staircase and pro ceeding as in Case (b) or Case (c) alwa ys maintains Prop erties 1 and 2. As neither Case (b) nor Case ( c) can b e applied an ymore, Prop erties 3 and 4 m ust hold as w ell. Now let us examine the graph G . Claim 8. The final gr aph G is a for est. Pr o of. W e pro v e b y induction a stronger stateme n t, that G will b e suc h a forest that the comp onen ts of the p oin ts b elow the staircase are disjoin t trees. When w e add an edge in Step (a), then the newly added po int that go es abov e the staircase will b e one o f t he endp oin ts, thus this prop erty is main ta ined. When w e add an edge in Step (b) or (c), then it connects t w o p oin ts b elow the staircase one of whic h w e immediately mo ve to the staircase, so w e are done b y induction. 6 V V 1 V 2 V 3 Figure 3: A t most 11 p o ints can b e in a mono c hromatic w edge Claim 9. A ny two-c olo rin g of G is a go o d c oloring of P . Pr o of. T ake an arbitrary tw o-coloring o f G . T ak e an arbitrary we dge V at time t that con ta ins at least 12 points of P t . If V con tains a point from abov e the staircase S t then b y Prop ert y 1 V con tains p oin ts of b oth colors. If V con tains at least 3 p oin ts from the staircase then V also con tains 3 consecutiv e points , thu s b y applying Prop erty 2 to the middle o ne V con tains b oth colors (as it contains b oth neighbors of this middle p oin t). F inally , if a w edge V does not con tain a p oint from ab ov e the staircase and con ta ins a t most 2 p oints f r o m the staircase then all the p oin ts b elo w the staircase that are cov ered b y V can b e co ve red b y 3 we dges con taining p o ints only from b elow the staircase (see the three w edges V 1 , V 2 and V 3 in Figure 3). By Prop ert y 4 eac h of these w edges cov er at most 3 points , th us V can contain altogether at most 11 p oints ( 2 from the staircase and 3 · 3 from b elow the staircase), a con tradiction. The ab ov e claim finishes the pro of of the theorem. 3 Miscellan y and a lo w er b ound W e ha ve seen in the In tro duction that if the po int set of The orem 5 is from the x + y + z = 0 plane, then the problem is equiv alen t to the co ve r-decomp osabilit y of homothetic copies of an equilateral triangle. Another special case is if the p oint set is on the x + y = 0 plane. The in tersection family of the o ctan ts with this plane is the family of b ottomless axis-p ar al lel r e ctangles ∗ . Bottomless rectangles were examined b y the first author [3] where it w as prov ed that any 3 -f old co verin g with b ottomless rectangles is decomposable in to tw o cov erings and a lso that an y finite po int se t can b e colored w ith t wo colors suc h that ev ery b ot to mless rec tangle con taining at least 4 po in ts con tains both colors. It w as also shown that these results are sharp. W e will use the ideas from [3] to pro ve the followin g claims, first of whic h is a strengthening of Theorem 5 in a sp ecial case and the se cond giving a sharp lo w er b ound for this sp ecial case, whic h also holds for the g eneral case. ∗ A set is a b ottomless axis-p ar al lel r e ctangle if it is the homothetic copy of the s et { ( x, y ) : 0 < x < 1 , y < 0 } . 7 1 2 3 4 5 6 7 8 9 10 (a) 1 2 3 4 5 6 7 8 9 10 T (b) Figure 4: Low er b o und constructions Claim 10. I f the pr oje ction of the original p oint set fr om R 3 onto the z = 0 plane yields a p oint set P having only p a i rw ise inc omp ar able p oints, then it admits a two-c oloring s uch that any tr anslate of a given o ctant that c ontains a t le ast 4 p oi nts c o ntains p oints with b oth c olors. Pr o of. W e use the same not a tions as in Section 2 . No w at an y time the p oin ts o f P t are pairwise incomparable. Order them according to their x co ordinate. W e will main tain a partia l coloring suc h that at an y time t : 1. Th ere a re no t w o consecutiv e p oints in this o rder that are not colored. 2. Th e colored p oin ts are colored a lternatingly . W e s tart with the empt y se t and then in a general step we add the p oint p t . If in the order it go es betw een t w o colored p oin ts then we leav e it uncolored. If it comes next to an uncolored p oin t then w e color these tw o p oints main ta ining the alternating color ing. At the end w e color the remaining uncolored p oints arbitrarily . W e claim that at any time t an y w edge cov ering at least 4 p oints co ve rs points from both colors already at step t of the coloring. Indeed, any w edge co vers consecutiv e po in ts and it cov ers at least 2 ( consecutiv e) colored p oin ts by Prop erty 1 and an y tw o consecutiv e colored p o ints are colored differen tly b y Prop ert y 2. Claim 11. F or any o ctant ther e exists a 10 p o i n t set P ⊂ R 3 such that its p r o j e ction onto the z = 0 plane yields a p o i n t s e t h a ving only p airwise inc omp a r able p oints, yet in an y two- c oloring of P ther e exists a tr anslate of a given o ctant that c ontains 3 p oints with the same c o lor and no other p oints. Pr o of. The p oint set on Figure 4(a ) has the needed prop erties (for simplicit y , the pro jection of the p oin t having the t th biggest z coo rdinate is denoted by t instead of p t ). Indeed, supp ose o n 8 the con t ra ry tha t there exists a t wo-coloring with no mono ch romatic w edge cov ering exactly 3 p oin ts. It is easy to c heck that the triple s (1 , 2 , 3) , (1 , 2 , 4) , (1 , 2 , 5 ) , ( 3 , 4 , 5) , (6 , 2 , 5) , (6 , 2 , 7 ) , (6 , 2 , 8) , ( 5 , 7 , 8) , (6 , 1 , 2) , (6 , 1 , 9) , (6 , 1 , 10 ) , (2 , 9 , 10) can all be co v ered b y some w edge at some time t . By the pigeonhole princ iple there are t w o p oints from (1 , 2 , 6) that ha v e the same colors. If e.g. 1 and 2 are colored red, then b y the first three triples 3 , 4 and 5 all m ust b e colored blue, but then the fourth triple is mono c hromatic, a contradiction. The analysis is similar if 2 a nd 6 ha ve the same color. Finally , if 1 and 6 is e.g. red and 2 is blue, then w e obtain a con tradiction from the last three triples, a s 9 a nd 10 should b e b oth blue b ecause of the p enultim ate and an tep en ultimate triples, but then the ultimate triple is mono c hromatic. This cons truction can b e mo dified a bit to impl y the same result fo r translates of a giv en triangle. Claim 12. T her e exists a 10 p oint set P ⊂ R 2 and a given triangl e T such that in any two- c oloring of P ther e exists a tr anslate of T that c ontains 3 p oin ts of the same c olor and no other p oints. Pr o of. The p oin t set and the triangle on Figure 4(b) has the needed properties, the pro of of this is exactly the same as o f t he previous claim. Finally w e note that this construction is a bit smaller then the one in [3], whic h had size 12 , s o w e obtain a smaller construction for that problem too b y taking the same points as in Claim 10 pro jected on to the y = 0 plane. 4 Lo cally finite co ve rings of the whole plane In this section we prov e Theorem 3. W e sa y that a cov ering is lo c al ly finite if ev ery compact se t in tersects only a finite n um b er of co verin g sets, i.e. homothetic copies of the giv en t ria ngle, in our case. In this section w e pro v e that any lo cally finite, 12 -fold co v ering of t he whole plane with homothetic copies of a triangle is decomp osable in to tw o co v erings. After an affine transformation w e can supp ose that the triangle is equilateral, w e will denote it b y T . W e will use Lemma 13. [Kö n ig’s I nfinity L emma, [4]] L et V 0 , V 1 , .. b e an infinite se quenc e of disj o int non- empty finite sets, and let G b e a gr ap h o n their union. Assume that every vertex v n in a set V n with n ≥ 1 has a neighb our f ( v n ) in V n − 1 . T h en G c ontains an infinite p ath, v 0 v 1 ... with v n ∈ V n for a l l n . T ake K 1 ⊂ K 2 ⊂ . . . compact sets suc h that their union is the whole plane. Let eac h v n b e a p ossible coloring of tho se finitely many triangles that in tersect K n suc h that eve ry p oint of K n is cov ered b y both colors. In this case V n is non-empty b ecause of Corollary 2 . The function f is the natural restriction to t he triangles that inters ect K n − 1 . The infinite path g ives a partition to t w o co v erings.  9 Remarks and ac kno wl edgmen t W e w ould lik e to thank Jean Cardinal for calling our a tten tion to this problem. References [1] B. Á cs: Síkfedések szétb on thatósága, Master Thesis (in Hungarian). [2] Jean Cardinal, p ersonal comm unication. [3] B. Keszegh, W eak conflict-free colorings of p o int sets and simple regions, The 19th Canadian Confer enc e on Computational Ge ometry (CCCG07 ) , Pro ceedings (2007), 97–100. [4] D. K önig: Theorie der Endlic hen und Unendlic hen Graphen, Kom binatorisc he T op ologie der Strec ke nk o mplexe, Leipzig, Ak ad. V erlag. [5] D. P álvölgyi, Dec omp osition of Geome tric Set Systems and Graphs, PhD t hesis, h ttp:// a rxiv.org/abs/1009.4641 [6] J. Pac h, D. P álv ö lgyi, G. Tóth, Surv ey on the Decomposition o f Multiple Cov erings, man uscript. [7] J. Pac h, G . T ardos, G. Tóth, Indecomp osable cov erings, Canadian Mathe matic al Bul letin 52 (2009), 451–46 3. [8] G . T ardos, G. Tóth, Multiple cov erings of the plane with triangles, D iscr ete and Computational Ge ometry 38 (20 07), 443–450 . 10