Colouring set families without monochromatic k-chains

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1 Colourig set families without moochromatic k-chais Shagik Das Roma Glebov Bey Sudakov Tua Tra April 17, 2018 Abstract A coloured versio of classic extremal problems dates back to Erdős ad Rothschild, who i 1974 asked which -vertex graph has the maximum umber of 2-edge-colourigs without moochromatic triagles. They cojectured that the aswer is simply give by the largest triagle-free graph. Sice the, this ew class of coloured extremal problems has bee extesively studied by various researchers. I this paper we pursue the Erdős Rothschild versios of Sperer s Theorem, the classic result i extremal set theory o the size of the largest atichai i the Boolea lattice, ad Erdős extesio to k-chai-free families. Give a family F of subsets of [], we defie a r, k)-colourig of F to be a r-colourig of the sets without ay moochromatic k-chais F 1 F 2... F k. We prove that for sufficietly large i terms of k, the largest k-chai-free families also maximise the umber of 2, k)-colourigs. We also show that the middle level, [], maximises the umber of 3, 2)-colourigs, ad give asymptotic results o the maximum possible umber of r, k)-colourigs wheever rk 1) is divisible by three. 1 Itroductio Sperer s Theorem o the size of the largest atichai i the Boolea lattice, which dates back to 1928, is oe of the fudametal theorems i extremal set theory. A atichai is a family of sets F 2 [] where o set is cotaied i aother; that is, F 1 F 2 for all distict F 1, F 2 F. Sperer [27] proved that 2 [] does ot have ay atichais with more tha sets, a boud that is easily see to be tight by cosiderig the family of all sets of size. Sice its iceptio, Sperer s Theorem has ispired a great deal of further research, as various extesios have bee prove. For istace, Erdős [6] determied the largest family F 2 [] without a k-chai; that is, without sets F 1 F 2... F k. Sperer s Theorem correspods to the case k = 2, ad Erdős exteded this by showig oe ought to take the k 1 largest uiform levels i 2 []. Whe k is odd, there are two isomorphic extremal families, ad we shall ot distiguish betwee them i this paper. I this paper we solve some istaces of the Erdős Rothschild extesios of these theorems. Before presetig our ew results, we first itroduce the class of Erdős Rothschild problems, which are multicoloured versios of classic extremal problems. Freie Uiversität Berli, Istitut für Mathematik, Arimallee 3, Berli, Germay. shagik@mi.fu-berli.de. School of Computer Sciece ad Egieerig, Hebrew Uiversity, Jerusalem , Israel. roma.l.glebov@gmail.com. Research supported by the ERC grat High-dimesioal combiatorics at the Hebrew Uiversity. Departmet of Mathematics, ETH, 8092 Zurich, Switzerlad. bejami.sudakov@math.ethz.ch. Research supported i part by SNSF grat Departmet of Mathematics, ETH, 8092 Zurich, Switzerlad. mah.tra@math.ethz.ch. Research supported by the Alexader Humboldt Foudatio. 1

2 1.1 Erdős Rothschild problems Oe of the earliest results i extremal combiatorics was obtaied i 1907, whe Matel [21] showed that ay -vertex graph with more tha 2 /4 edges must cotai a triagle, a boud that the complete balaced bipartite graph shows to be best possible. This was later exteded by Turá [28], who determied the largest umber of edges a -vertex K t -free graph ca have, for ay t. More geerally, the Turá umber of a graph H, deoted ex, H), is the maximum umber of edges i a H-free graph o vertices. Various extesios ad variatios of the Turá problem have bee pursued through the years. Oe such problem was proposed by Erdős ad Rothschild [7] i They asked which -vertex graph had the maximum umber of two-edge-colourigs without moochromatic triagles. Clearly, sice the complete balaced bipartite graph is itself triagle-free, ay twocolourig of its edges is moochromatic-triagle-free, ad hece oe ca have at least 2 2 /4 such colourigs. I order to do better, oe would have to take a graph with more edges, which, by Matel s Theorem, implies the existece of triagles. These triagles impose restrictios o the twocolourigs, as their edges caot be coloured moochromatically. Erdős ad Rothschild believed that the restrictios from the triagles would more tha couteract the extra possibilities offered by the additioal edges, ad cojectured i [7] that 2 2 /4 is i fact the best oe ca do. It was some twety years before Yuster [30] proved the cojecture for sufficietly large. There are some obvious geeralisatios of this problem of Erdős ad Rothschild oe may ask it for graphs other tha the triagle, ad oe may icrease the umber of colours used. Let a r, F )-colourig of a graph G be a r-colourig of its edges without ay moochromatic copies of F. The questio is the to determie which -vertex graphs G maximise the umber of r, F )-colourigs. Oce agai, a atural lower boud is obtaied by cosiderig all r-edge colourigs of the largest F -free graph, which shows that the maximum umber of r, F )-colourigs of a -vertex graph is at least r ex,f ). I 2004, Alo, Balogh, Keevash ad Sudakov [1] greatly exteded Yuster s result by showig this lower boud was tight wheever F = K t for t 3 ad r {2, 3}. Iterestigly, they further demostrated that this was ot the case whe oe has four or more colours, providig some better bouds i this rage. Exact results were later obtaied by Pikhurko ad Yilma [25], who determied which -vertex graphs maximise the umber of 4, K 3 )- ad 4, K 4 )-colourigs. Pikhurko, Stade ad Yilma [24] have recetly itroduced a asymmetric versio of this problem, reducig its log-asymptotic solutio to a large but fiite optimisatio problem. I recet years, there have bee a variety of papers studyig the Erdős Rothschild problem i various settigs. Lefma, Perso, Rödl ad Schacht [17] studied the problem i the settig of three-uiform hypergraphs, with moochromatic copies of the Fao plae forbidde, before Lefma, Perso ad Schacht [18] cosidered arbitrary k-uiform hypergraphs. Further results alog this lie of research ca be foud i [10, 11, 16]. Movig the problem ito the domai of extremal set theory, Hoppe, Kohayakawa ad Lefma [13] solved the Erdős Rothschild extesio of the famous Erdős Ko Rado Theorem [8]. Hoppe, Lefma ad Oderma [14] provided iitial results for the vector space aalogue of the Erdős Ko Rado Theorem, before Clemes, Das ad Tra [3] preseted a uified proof extedig some of these results. I the cotext of additive combiatorics, the Erdős Rothschild extesio for sum-free sets has bee pursued by Hà ad Jiméez [9] for abelia groups ad Liu, Sharifzadeh ad Stade [19] for subsets of the itegers. 2

3 1.2 Our results Followig their work o itersectig vector spaces, Hoppe, Lefma ad Oderma [14] suggested the ivestigatio of Erdős Rothschild problems i the cotext of the power set lattice. Sperer s Theorem o atichais is arguably the most importat extremal result i this settig, ad we cosider the correspodig Erdős Rothschild extesio, as well as that of Erdős s result o k-chai-free families. To this ed, give a set family F 2 [], we defie a r, k)-colourig of F to be a r- colourig of the sets i F without ay moochromatic k-chais F 1 F 2... F k. Give r 2 ad k 2, the goal is to determie which families have the maximum possible umber of r, k)-colourigs. Let us defie fr, k; ) to be this maximum. Let us first cosider the case k = 2. Here we forbid a moochromatic 2-chai or, i other words, require that each colour class be a atichai. As before, ay r-colourig of a atichai is a r, 2)-colourig. By Sperer s Theorem [27], the largest atichai i 2 [] has size. The lower boud fr, 2; ) r thus follows, ad we seek to determie if this is best possible. This is somewhat trivial whe r = 2. Ideed, let F 2 [] be a family maximisig the umber of 2, 2)-colourigs, ad let F 0 F be the subfamily of all miimal sets i F. Observe that ay 2, 2)-colourig of F is determied by the colourig of F 0, sice ay set F F \ F 0 cotais a set i F 0, ad must thus be oppositely-coloured. Hece F ca have at most 2 F 0 2, 2)-colourigs, ad sice F 0 is a atichai, we have F 0. Thus 1) is tight for r = 2, ad it is ot hard to show that oe oly has equality for the largest atichais; that is, whe F is oe of) the middle layers) of the Boolea lattice. To show that the lower boud remais tight for r = 3 requires cosiderably more work, ad is the cotet of our first theorem. Theorem 1.1. There is some 0 N such that for every iteger 0, f3, 2; ) = 3 ; that is, every family F 2 [] has at most 3 3, 2)-colourigs. Moreover, we have equality if ad oly if F = [] ) or F = [] /2 ). Just as for the Erdős Rothschild extesio for Turá s Theorem, we caot expect equality i 1) to hold for larger r. Ideed, for r = 4, suppose = 2m + 1, ad cosider the family F = ) [] m [] ) m+1. Partitio the four colours ito two pairs, ad colour the sets i [] m) with oe of the pairs, ad those i [] m+1) with the other. Each such colourig, of which there are 2 m) 2 m+1) = 4, is clearly a 4, 2)-colourig. This matches the lower boud of 1). However, as there are six ways to partitio the colours, ad these give rise to almost disjoit sets of 4, 2)-colourigs, we fid that F has more 4, 2)-colourigs tha the largest atichai []. For larger values of r, this family F has expoetially more r, 2)-colourigs tha the lower boud give above. We ow tur to the case whe k 3. Recall that Erdős [6] proved that the largest k- chai-free family i 2 [] cosists of the k 1 largest uiform levels; that is, the collectio of all sets whose sizes lie betwee k + 2)/2 ad + k 2)/2. Let the size of this family be deoted by m k 1, so that m k 1 = +k 2)/2 i= k+2)/2 i). Sice every r-colourig of this family is a r, k)-colourig, we have the iequality 1) fr, k; ) r m k 1. 2) The case k 3 appears to be rather more complicated tha k = 2. Ideed, it was trivial 3

4 to boud the umber of 2, 2)-colourigs that a family could have. Whe r = 2 ad k 3, the lower boud of 2) is agai sharp, but our proof is much more ivolved, ad forms our ext result. Theorem 1.2. There exists a absolute costat C > 0 such that for itegers ad k with k 2 ad Ck 4 log k, f2, k; ) = 2 m k 1; that is, the umber of 2, k)-colourigs of a family F 2 [] is at most 2 m k 1. Moreover, we have equality if ad oly if F is a k-chai-free family of maximum size. Fially, it is fairly straightforward to see that the lower boud i 2) is ot best possible for larger values of r; ideed, for r 5, oe ca agai do expoetially better. Our fial result uses the machiery of hypergraph cotaiers to provide upper bouds o fr, k; ). These determie fr, k; ) log-asymptotically for more tha half of pairs r, k). Propositio 1.3. For k 2, r 3 ad ε > 0 there exists 0 r, k, ε) such that for 0 r, k, ε) we have fr, k; ) 3 1 rk 1+ε) 3. Moreover, if rk 1) is divisible by three, the as well. 1.3 Orgaisatio ad otatio fr, k; ) rk 1 ε) The remaider of this paper is orgaised as follows. I Sectio 2 we prove Theorem 1.1, ad the prove Theorem 1.2 i Sectio 3. Propositio 1.3 is proved i Sectio 4. We remark that these sectios are idepedet of oe aother, ad ca be read i ay order. We close the paper with some cocludig remarks i Sectio 5. Oe otio we shall use throughout is that of a comparability graph. Give a family F of subsets of [], the comparability graph of F is the graph GF) whose vertices are the sets i F, with a edge {F 1, F 2 } wheever F 1 F 2. For ay set F [], the up-degree respectively dow-degree) of F i F, deoted by d + F, F) respectively d F, F)), is the umber of sets F i F such that F F respectively F F ). The degree of F i F, deoted by df, F), is the sum of its up-degree ad dow-degree i F, ad couts the sets i F comparable to F excludig F itself, if F F). We deote by N + F, F), N F, F) ad NF, F) the subfamilies of up-, dow- ad all eighbours of a set F i the family F. That apart, we make use of stadard combiatorial otatio. We deote by [] the set {1, 2,..., }, ad shall take that to be the groud set for our set families. Give a set X ad some k N, ) X k is the family of all k-subsets of X, while 2 X is the family of all subsets of X, regardless of size. Fially, uless stated otherwise, all logarithms are biary. 2 Three-colourigs without moochromatic two-chais I this sectio, we will show that the umber of 3, 2)-colourigs of a set family is maximised oly by the largest atichais; that is, by the middle levels, [] ) or [] /2 ). Theorem 1.1. There is some 0 N such that for every iteger 0, f3, 2; ) = 3 ; that is, every family F 2 [] has at most 3 3, 2)-colourigs. Moreover, we have equality if ad oly if F = [] ) or F = [] /2 ). 4

5 Before delvig ito the details of the proof, we provide a heuristic argumet for why the largest atichais should also maximise the umber of 3, 2)-colourigs. I a atichai, a 3, 2)-colourig ca be formed by arbitrarily assigig each of the sets oe of the three colours. Therefore, i order to have more 3, 2)-colourigs, oe must cosider a larger family. By Sperer s Theorem, ay larger family must cotai comparable pairs. These pairs place restrictios o our colourigs: oce the colour of a give set is fixed, ay comparable sets have at most two of the three colours available to them. To overcome these restrictios, the family must i fact be sigificatly larger tha the largest atichais. However, oe ca tha show that the family must cotai may sets that are comparable to a large umber of other sets. This i tur leads to eve stroger restrictios o the 3, 2)- colourigs of the family. Ideed, cosider such a set F, ad let G be the sets i the family to which F is comparable. I a typical colourig, we would expect to see at least two colours appearig i G, i which case the colour of F is determied, rederig this set redudat. I order for F to actually icrease the umber of 3, 2)-colourigs, we would eed G to be moochromatic, which is a very restrictive coditio. It thus appears ulikely that larger families could have more 3, 2)-colourigs. I what follows, we formalise this argumet ad show this ituitio to be true. Proof. Let F 2 [] maximise the umber of 3, 2)-colourigs ad let C be the set of all 3, 2)- colourigs of F. The lower boud of 1) implies C 3. 3) We must establish a matchig upper boud o C. We begi by deducig some structural iformatio about F. Suppose F cotais a 3-chai F 1 F 2 F 3. If ay two of F 1, F 2 ad F 3 shared the same colour, they would form a moochromatic 2-chai. Hece i every colourig i C, F 1, F 2 ad F 3 must receive pairwise differet colours, ad thus the colour of F 3 is determied by those of F 1 ad F 2. It follows that F \ {F 3 } has at least as may 3, 2)-colourigs as F. We therefore may assume that F is 3-chai-free, ad ca thus be partitioed ito two atichais by Mirsky s theorem [23]. 1 I other words, the comparability graph GF) of F, as defied i Sectio 1.3, is bipartite. We deote by I the subfamily of F cosistig of isolated vertices of GF); that is, those sets i F that are icomparable to all other sets i F. Next, we iteratively remove from the remaider F \ I sets of degree at most i the iduced subgraph of GF), ad call the subfamily of these sparse removed sets S. We are left with a bipartite subgraph of miimum degree at least, ad call the subfamilies correspodig to the two idepedet sets A ad B. Hece da, B) for every A A, ad db, A) for every B B. Give this structure, our goal is to show that a relatively small umber of sets i F are cotaied i eough comparable pairs to limit the umber of 3, 2)-colourigs of F. To this ed, we ow defie some otatio we will use throughout this proof. Defiitio 2.1. Fix a subfamily X B ad a 3, 2)-colourig ϕ C. We write B ϕ mc for the subfamily of B cosistig of vertices whose eighbourhoods i A are moochromatic uder ϕ. Let A 3 = {F A : df, X ) = 0}, A 2,ϕ = {F A : df, B ϕ mc), df, X B ϕ mc) = 0} ad A 1,ϕ = {F A : df, B ϕ mc) < }. Observe that A 3 is ot ecessarily disjoit from A 2,ϕ A 1,ϕ. Observe that B ϕ mc ad A 1,ϕ deped oly o ϕ, while A 3 depeds oly o X. The followig claim, to be prove at the ed of this sectio, is the key of the proof. 1 This does ot affect our claims of uiqueess; if we started with a family F that was ot the middle level of the Boolea lattice, the 3-chai-free subfamily would also ot be the middle level, as it would still have comparable pairs. 5

6 Claim 2.2. For sufficietly large, there exist families X B ad C C such that i) X 0.01 B, ii) C 9 10 C, ad iii) max{ A 3, A 2,ϕ } 0.01 A for all colourigs ϕ i C. Before cotiuig, let us remark that, by 3) ad Claim 2.2 ii), the size of C is at least C 9 10 C ) With X as i Claim 2.2, we collect the followig estimates, whose proofs are also deferred to the ed of this sectio. Claim 2.3. For sufficietly large, the followig iequalities hold: a) max{ A, B } I, b) S 1.01 ) ) I, ad c) A 2,ϕ A 1,ϕ + Bmc ϕ 1.02 ) ) I S for every ϕ C. Assumig Claims 2.2 ad 2.3, we shall provide the desired upper boud o C. Ideed, we first appeal to Claim 2.2 to obtai a subset of vertices X B ad a subfamily of colourigs C C with properties i) iii). We ow show that for a arbitrary colourig ϕ i C, it is possible to determie the colourig completely by askig a small umber of questios with a bouded umber of possible aswers. This will show that, uless F is a middle level of the Boolea lattice, C is smaller tha the lower boud i 4), thus provig that the middle levels are the oly families with the maximum possible umber of 3, 2)-colourigs. First, for every x X, we ask for the colour of x, ad also whether its eighbourhood i A is moochromatic ad, if it is, which colour it has. There are 9 possible aswers for each vertex x X, ad so by Claims 2.2 i) ad 2.3 a), it follows that the total umber of aswers at this ) stage is ot greater tha 9 X B I. The aswers to these questios restrict the possible colours assiged to sets i A. Those foud to be i a moochromatic eighbourhood of a vertex i X have their colour determied. Those with a eighbour i X have at most two colours left as they caot have the same colour as their eighbour). Oly those sets i A without a eighbour i X could still have three colours available. Thus we have a partitio A = C 1 C 2 C 3, where C i cosists of all sets F A such that there are exactly i colours available for F, leavig at most 3 C3 2 C2 ways for the sets i A to be coloured. After colourig the vertices of A, we ca idetify the family B mc B of all vertices i B whose eighbourhoods i A are moochromatic. Observe that each elemet of B mc ca receive 2 colours, while vertices i B \ X B mc ) oly have oe colour available. It follows that the umber of possibilities of extedig the colourig to B is 2 Bmc. Notice that B mc = Bmc, ϕ ad C 3 = A 3. Moreover, C 2 A 2,ϕ A 1,ϕ, because every elemet F C 2 \ A 1,ϕ has at least eighbours i Bmc ϕ as F / A 1,ϕ ) ad o eighbours i X Bmc ϕ otherwise the colour of F is already determied). Claims 2.2 iii), 2.3 a) ad 2.3 c) thus force C ) ) I ad C 2 + B mc 1.02 ) ) I S. 6

7 Therefore, give the aswers for the sets i X, there are at most 3 C 3 2 C 2 2 Bmc ) I 2 S possibilities to fiish the colourig of A B. We proceed to boud the umber of colourigs of S. Let {{F 1, G 1 },..., {F t, G t }} deote a maximal matchig i the comparability graph GS). Sice F i ad G i are comparable for each 1 i t, there are at most 6 ways to colour each pair {F i, G i }, ad hece the umber of possible colourigs for the matchig is at most 6 t. O the other had, it follows from the defiitio of I ad the maximality of the matchig that every set F i S ot i the matchig is adjacet to some previously-coloured set i A B {F 1, G 1,..., F s, G s }, ad therefore has at most two available colours. The family S ca therefore be coloured i at most 6 t 2 S 2t possible ways, which is at most 6 S sice t S /2. Fially, the umber of ways to colour I is 3 I. Puttig these iequalities together, we ca upper boud the umber of colourigs i C by C ) I ) I 2 S 6 S 3 I = ) I ) S 6/ ) I ) 1.01 I 6/ I, 5) where the third iequality follows from Claim 2.3 b), ad the last holds sice I due to Sperer s Theorem. Combiig 4) ad 5), we fid that I 0. Hece I must be a atichai of size, ad therefore oe of the middle levels of 2 []. Sice I is the set of isolated vertices i the comparability graph GF), ad ay other set is comparable to some sets i the middle levels, this forces F = I, completig the proof. It remais to prove Claims 2.2 ad 2.3, a task we ow begi. Proof of Claim 2.2. If A =, the X = trivially has the desired properties. We thus assume A. By the defiitio of A ad B, we must have B. Observe that for every colourig ϕ, the set Bmc ϕ is well-defied ad idepedet of the radom choice of X that we shall ow make. Let X B) p deote the radom subfamily of B, where each set i B is icluded idepedetly with probability p = 1/ log. Assumig is sufficietly large ad applyig the Cheroff boud, we have P X > 0.01 B ) exp 1/3). 6) For a 3, 2)-colourig ϕ of F, let E ϕ be the evet that A 2,ϕ A 3 > 0.01 A. We will show it is ulikely that this occurs for may colourigs simultaeously. More precisely, P {ϕ C : E ϕ } > 1 ) 10 C 1000 exp 1/3). 7) ) Clearly, 6) ad 7) together imply the existece of X B ad C properties. C with the desired 7

8 It thus remais to show that 7) holds. As every elemet i A 3 has at least eighbours i B, oe of which are selected i the radom subfamily X, the uio boud gives E[ A 3 ] A 1 p) A exp p ). Similarly, sice A 2,ϕ has eighbours i B ϕ mc \ X, we fid E[ A 2,ϕ ] A 1 p) A exp p ). Combiig these bouds with Markov s iequality, we obtai PE ϕ ) = P A 2,ϕ A 3 > 0.01 A ) 2 A exp p ) 0.01 A 100 exp 1/3). Liearity of expectatio the gives E {ϕ C : E ϕ } ) = ϕ C PE ϕ) 100 exp 1/3) C, ad hece aother applicatio of Markov s iequality implies P {ϕ C : E ϕ } > 1 ) 10 C 100 exp 1/3) C = 1000 exp 1/3). C /10 This fiishes our proof of Claim 2.2. We coclude this sectio with a proof of Claim 2.3, for which we shall use the followig special case of a result of Kleitma [15]. This provides a lower boud o the umber of comparable pairs i a large set family of a give size see also [5], which characterises the extremal families). Theorem 2.4 Kleitma [15]). A subfamily of 2 [] with + t sets must cotai at least t comparable pairs Proof of Claim 2.3. Property a) follows immediately from Sperer s Theorem after otig that I A ad I B are idepedet sets i GF), ad therefore atichais. Sice I cosists of isolated sets i GF), ad S was formed by successively removig sets of degree at most, it follows that the comparability graph GI S) has at most S ) + 2 edges. By Theorem 2.4, this forces I S S, ad cosequetly, oe has S 1.01 ) ) I for sufficietly large, establishig b). We fially prove c). Let F = I S A 1,ϕ Bmc. ϕ By the defiitio of S, there are at most S edges icidet to S i GF ). Moreover, every vertex from A 1,ϕ is icidet to at most vertices from B ϕ mc. As A ad B are idepedet sets, ad I is the set of isolated vertices, there are o other edges i GF ). I total, GF ) has at most S + A 1,ϕ ) edges. We thus get F ) + 2 S + A 1,ϕ ) by applyig Theorem 2.4. This implies A 1,ϕ + Bmc ϕ 1.01 ) ) I S, where we have assumed is sufficietly large. Moreover, A 2,ϕ 0.01 A 0.01 ) ) I by Claims 2.2 iii) ad 2.3 a). Combiig these iequalities, we coclude completig the proof. A 2,ϕ + A 1,ϕ + B ϕ mc 1.02 ) ) I S, 8

9 3 Two-colourigs without moochromatic k-chais I this sectio we prove Theorem 1.2, which we first restate below. Theorem 1.2. There exists a absolute costat C > 0 such that for itegers ad k with k 2 ad Ck 4 log k, f2, k; ) = 2 m k 1; that is, the umber of 2, k)-colourigs of a family F 2 [] is at most 2 m k 1. Moreover, we have equality if ad oly if F is a k-chai-free family of maximum size. Recall that this shows the lower boud i 2) is tight. The largest k-chai-free families, by a result of Erdős [6], cosist of the k 1 largest levels of the Boolea lattice, ad therefore have size m k 1 = +k 2)/2 i= k+2)/2 i). Ay larger family would have to cotai k-chais, ad we eed to show that these chais place too may restrictios to allow for a larger umber of 2, k)-colourigs. Just as we did i the proof of Theorem 1.1, we shall do this by partitioig ay cadidate family i such a way that eables us to boud the umber of 2, k)-colourigs effectively. However, i this istace, the partitio we use is much more complex, ad we describe it i the followig propositio. For coveiece, we shall use the parameters ε = 1/500k 2 ) ad ω = 4k log1/ε)/ε. Propositio 3.1. If F 2 [] is a family with at least oe 2, k)-colourig, the F admits a partitio F = A U D P R, where each part is further partitioed ito k 1 subparts e.g. A = k 1 i=1 A i, ad similarly for U, D, P ad R), that satisfies the followig properties. P1) F has at most 2 A + U + D +ε R P 2, k)-colourigs. P2) For 1 i k 1, A i is a atichai. P3) If 1 i k 1 ad F is a set i oe of U i, U i+1, D i, D i 1, P i or R i where we take U k = D 0 = ), the F is comparable to at most 2ω sets i A i. Moreover, U 1 = D k 1 =. P4) For each B i {U i D i P i, U i D i 1, D i U i+1 } where, agai, U k = D 0 = ), the family A i B i has at most 3ω B i comparable pairs. P5) For ay subfamily H F, there is a atichai H H of size H H /2k 2). As ca be see from the property P1) above, this partitio gives us some cotrol over the umber of 2, k)-colourigs of the set family F. I the ext subsectio, we shall show how oe may combie this with the other properties guarateed by Propositio 3.1 to prove Theorem 1.2, thus motivatig this complex partitio. Subsectio iformally explais how the partitio will be created, before the proof of Propositio 3.1 is give i Subsectio 3.2. The fial subsectio is devoted to the proofs of some techical lemmata we shall require. 3.1 Coutig the colourigs I this subsectio we will show how the partitio from Propositio 3.1 implies Theorem 1.2. Observe that A, which by property P2) is the uio of k 1 atichais, is k-chai-free, ad hece by the theorem of Erdős [6] has size at most m k 1. Ideed, i the extremal cofiguratio, which is the uio of the k 1 largest levels of the Boolea lattice, each A i is oe of the uiform levels, while we have U = D = P = R =. 9

10 By property P1), if F is a family with more 2, k)-colourigs, the at least oe of U, D, P or R must be o-empty. We shall the use properties P3) ad P4), which boud the umber of comparable pairs i certai subfamilies, to obtai upper bouds o the sizes of the parts of the partitio, which will i tur show that F has strictly fewer tha 2 m k 1 comparable pairs. This requires the use of supersaturatio results: we shall have to deduce from the small umbers of comparable pairs that the correspodig families are small. Note that Theorem 2.4 is such a result, showig that a family with few comparable pairs caot be much larger tha the middle level of the Boolea lattice. While that result is tight, our partitio separates ito k 1 levels, ad they caot all occupy the full middle level. Hece we will have to derive a weighted versio of the supersaturatio result that is still tight for multiple levels close to the middle level). We thus defie the k-)weight of a set F 2 [] to be w k F ) = mi { ) 1, F k 2 ) } 1, ad the weight of a set family as the sum of the weights of its members, w k F) = F F w kf ). Note that the weight of a set icreases with the distace of the set to the middle level, but this icrease is capped to prevet udue ifluece beig give to sets that are much smaller or larger tha what we expect to fid i the optimal costructio. With this otatio i place, we preset our supersaturatio lemma, which we shall prove i Subsectio 3.3. Lemma 3.2. There is some costat C > 1 such that the followig statemet holds for all δ 0, 1 2) ad all itegers ad k with k 2 ad Cδ 3 k 2. If F is a subfamily of 2 [] with w k F) 1 + r 1 for some r R, the the umber of comparable pairs i F is at least 1 2 δ) r. Usig Lemma 3.2, it follows that the families i property P4) of Propositio 3.1 have small weight. However, i order to apply property P1) to boud the umber of 2, k)-colourigs, we will have to cotrol the sizes of these families istead. The followig lemma, whose proof is also i Subsectio 3.3, allows us to covert betwee weights ad sizes. Lemma 3.3. Let ad k be itegers with k 2 ad 4k 2. Suppose F 0, F 1,..., F s are subfamilies of 2 [] such that F 0 + s i=1 α i F i m k 1 + t for some positive reals α 1,..., α s, ad o-egative iteger t. The w k F 0 ) k2 ) s i=1 ) 1 α i w k F i ) k 1 + t. Armed with these lemmata, together with Propositio 3.1, we are i positio to prove our theorem. Proof of Theorem 1.2. Suppose Ck 4 log k, for some costat C large eough to satisfy the iequalities that will follow, ad that F 2 [] maximises the umber of 2, k)-colourigs. We apply Propositio 3.1 to obtai the claimed partitio of F for which the properties P1) P5) hold. I light of the lower boud of 2), the umber of 2, k)-colourigs of F must be at least 2 m k 1. By property P1), the umber of 2, k)-colourigs of F is at most 2 A + U + D +ε R P, ad thus m k 1 A + U + D + ε R + α P, 8) 10

11 where α = 1 2 log 3. Our first claim shows that the size of R ca be cotrolled by the sizes of U, D ad P. Claim 3.4. R 3k U + D + α P ). Proof. We start by applyig the property P5) to each subfamily R i, obtaiig a atichai R i R i. Defie R = i R i, ad ote that R R /2k 2). Further defie G i = A i R i, ad cosider the comparable pairs i G i. By P2) ad P5), both A i ad R i are atichais, ad hece the oly comparabilities i G i come from A i R i. By P3), there are at most 2ω R i such pairs. Applyig Lemma 3.2 with δ = 1 6 ad r = 6ω 1 R i, we deduce that w k G i ) 1 + 6ω 1 R i 1. Lettig G = i G i ad summig over i [k 1], we have w k G) k 1 + 6ω 1 R 1. Applyig Lemma 3.3 with s = 1, α 1 = 1, F 0 = G ad F 1 =, ad the usig 8), we fid G m k 1 + 6ω 1 R A + U + D + ε R + α P + 6ω 1 R. 9) Note that G = A R, ad thus G = A + R. Substitutig this ito 9) ad rearragig gives 1 6ω 1 ) R U + D + ε R + α P. Recallig that R R /2k 2) ad rearragig agai, we have the desired boud, sice ) 1 1 6ω 1 3k R ε R U + D + α P, 2k 2 where the first iequality follows from the facts that ε = 1/500k 2 ) ad 18ω. By combiig 8) with Claim 3.4, we ca lower boud the sizes of A, U, D ad P, obtaiig m k 1 A kε) U + D + α P ). We covert this ito a lower boud o the weights of these families by usig Lemma 3.3 with s = 3, F 0 = A, F 1 = U, F 2 = D, F 3 = P, α 1 = α 2 = 1 + 3kε, α 3 = 1 + 3k)α ad t = 0. This gives ) k 1 w k A) k kε) w k U) + w k D) + αw k P)) w k A) k ) w k U) + w k D)) w kp). 10) To complete the proof, we shall obtai a upper boud o the weights of these families as well, with the combiatio of the two oly beig satisfied whe U = D = P =. To this ed, for each 1 i k 1, let B i be the family from {U i D i P i, U i D i 1, D i U i+1 } that has greatest weight. Our ext claim shows that these weights caot be too small. Claim 3.5. i w kb i ) k ) wk U) + w k D)) w kp). Proof. Sice we could always have chose B i = U i D i P i, we must have w k B i ) w k U i ) + w k D i ) + w k P i ). Summig these iequalities over all i, we have k 1 w k B i ) w k U) + w k D) + w k P). 11) i=1 11

12 O the other had, cosider the families i {U i, D i : i [k 1]}, ad ote by P3) we have U 1 = D k 1 =. Suppose, for some j 2, U j has the greatest weight out of these families a similar argumet applies whe some D j, j k 2, is the heaviest). As the heaviest of the families, we must have w k U j ) 1 2k w ku) + w k D)). We ca agai boud the total weight of the families B i from below by the followig choices. For i j 1, take B i = D i U i+1, ad B i = U i D i P i whe i j. We would the have k 1 k 1 w k B i ) w k D i ) + w k U i )) + w k U i ) + w k D i ) + w k P i )) i=1 j 1 i=1 i=j k 1 k 2 w k U i ) + w k D i ) + w k U j ) = w k U) + w k D) + w k U j ). i=2 i= ) w k U) + w k D)). 12) 2k Takig a covex combiatio of these lower bouds, with coefficiet 9 for 12), we arrive at the claimed lower boud o i w kb i ). 10 for 11) ad 1 10 To fiish the proof, we ow boud the weights of the families B i from above. Usig P4), we observe that the umber of comparable pairs i the family A i B i is at most 3ω B i 3ω U i + U i+1 + D i + D i 1 + P i ). Applyig Lemma 3.2 with δ = 1 6 ad r = 9ω 1 U i + U i+1 + D i + D i 1 + P i ), we fid ) 1 w k A i B i ) 1 + 9ω 1 U i + U i+1 + D i + D i 1 + P i ) 1 + 9ω 1 w k U i ) + w k U i+1 ) + w k D i ) + w k D i 1 ) + w k P i )), where the secod iequality follows from the fact that every set i 2 [] has weight at least 1. Summig over 1 i k 1, k 1 k 1 w k A) + w k B i ) = w k A i B i ) k 1 + 9ω 1 2w k U) + 2w k D) + w k P)). 13) i=1 i=1 O the other had, we have the lower boud k 1 w k A) + w k B i ) w k A) ) 20k wk U) + w k D)) w kp) i=1 k k w ku) + w k D)) w kp), 14) where we use Claim 3.5 for the first iequality ad 10) for the secod. As > 1080ωk, 13) ad 14) ca oly be simultaeously satisfied whe w k U) = w k D) = w k P) = 0. This i tur implies U = D = P =. By Claim 3.4, it follows that R = as well. Hece F = A, the uio of k 1 atichais. Thus F is a k-chai-free family, which ca have size at most m k 1, with at most 2 m k 1 two-colourigs. This completes the proof of Theorem

13 3.2 Formig the partitio Now that we have see how the partitio from Propositio 3.1 ca be used to establish Theorem 1.2, we shall describe how a partitio with properties P1) P5) ca be formed. We begi with a iformal overview of the process, before providig a detailed proof of the propositio A overview For ispiratio, we first cosider the optimal costructio, which is the uio of the k 1 largest uiform levels of the Boolea lattice. This family admits a atural partitio ito k 1 atichais. Moreover, each set cotais may sets from the atichai below, ad is cotaied i may sets from the atichai above. We shall edeavour to build a similar structure a sequece of atichais, with each set cotaiig may sets from the atichai below, ad beig cotaied i may sets from the atichai above. This large degree i the comparability graph GF) as defied i Sectio 1.3) will be importat for us, as we will be able to use it to create may chais. If we fid a k 1)- chai F 1 F 2... F k 1 that is moochromatic i may colourigs, the it follows that ay sets that cotai F k 1 must all have the opposite colour. This allows us to remove all such sets ito a separate family R. As the colour of these sets is determied, we do ot lose may colourigs of the etire family F whe we do so. Similarly, ay sets that are cotaied i F 1 could also be removed. Sometimes, though, we may ecouter sets that do ot have the desired large degrees ito their eighbourig atichais. We take such sets out of their atichais, ad place them i separate families. If they are cotaied i too few sets from the atichai above, we call them up-sparse, ad place them i the family U. If they cotai too few sets from the atichai below, we ame them dow-sparse, ad place them i D. As we have see i the previous subsectio, their low degrees ito the evetual) atichais will allow us to exploit the supersaturatio result of Lemma 3.2. At this stage i the process, the parts A 1 up to A k 1 may ot actually be atichais, but could cotai a few comparable pairs. Our ext step is to remove these pairs to form geuie atichais. If a pair is ofte moochromatic, the we ca exted it ito a moochromatic chai, which agai allows us to remove a large umber of sets ito R without losig may colourigs. O the other had, if a pair is more ofte oppositely-coloured, the we place it i the family P istead. Note that for the pairs i P, we kow that out of the four ways the two sets could have bee coloured, two are much more likely to occur, which allows us to better boud the umber of colourigs with respect to the sets i P. This explais the key ideas behid the formatio of the partitio, as well as the roles played by the parts R, U, D ad P. Oce we have completed the steps described above, there will be some fial cleaig of the partitio to esure all the properties P1) P5) hold, after which the proof of Propositio 3.1 will be complete. We ow proceed to the details of the procedure The detailed procedure To begi, observe that if F cotaied a 2k 1)-chai C, the i ay two-colourig of F, C would cotai a moochromatic k-chai. Hece, if F admits eve a sigle 2, k)-colourig, we deduce that F must be 2k 1)-chai-free. Appealig to Mirsky s theorem [23], we see that F ca be partitioed ito 2k 2 atichais. Explicitly, let A i be the maximal elemets i the poset F \ i 1 j=1 A j ), which gives a partitio F = A 1... A 2k 2 15) 13

14 ito 2k 2 atichais some of which may be empty). This partitio has the followig quality. Q1) If i < j, F A i ad G A j, the F G. Q1) obviously holds for the partitio described above. However, our partitio shall be dyamic, as we will move sets betwee parts to reach the desired fial partitio. We shall esure Q1) is maitaied throughout the process. It will also be coveiet to fix a liear extesio F, ), such that give sets F, G F, we have F G oly if F G. We further require that the liear extesio start with all sets i A 2k 2, followed by those i A 2k 3, ad so o, util the sets i A 1 are listed last. As we proceed, we will deote by C the set of 2, k)-colourigs of F curretly uder cosideratio. To begi with, C will cotai all 2, k)-colourigs of F. However, we shall occasioally colour some sets i F, ad will oly retai i C those colourigs that agree with the partial colourig of F. Oe of the ways we shall colour sets is through the brachig operatio, which we will ow describe. The mai idea is to build chais util we fid a large umber of sets that must ofte be moochromatic, thus allowig us to colour ad remove may sets while oly shrikig C moderately. 2 Brachig from a coloured vertex: Recall that ε = 1/500k 2 ) ad ω = 4k log1/ε)/ε. Assume some set A F has bee coloured, say red, ad there is some idex l such that d + A, A l ) > ω or d A, A l ) > ω). Because j ε ) 2 εj = 1, oe of the two followig statemets must hold: i) I at least a 1 2 ε )-fractio of the colourigs i C, all sets i N + A, A l ) respectively, N A, A l )) are all coloured blue. Note that this correspods to the j = 0 summad above. ii) There exists a iteger j 1 such that i at least a 1 2 ε )2 εj -fractio of all colourigs i C, the jth set with respect to i N + A, A l ) respectively, N A, A l )), say B, is the first red set i the eighbourhood. I the first case, we ca colour all the sets i N + A, A l ) respectively, N A, A l )) blue, remove them from A l ad place them i R, ad restrict C to the 1 2 ε ) C colourigs where these sets are all blue. I the latter case, we colour B red ad the precedig sets i N + A, A l ) respectively, N A, A l )) blue, remove these sets to R, ad restrict C to the remaiig 1 2 ε )2 εj colourigs. We ca piece together these brachig steps ito the followig operatios. Brachig up: Let A be a coloured set i F, ad let l k 1 be a iteger such that d + A, A l ) > ω ad d + F, A i 1 ) > ω for all 2 i l ad F A i. We brach from A to its supersets i A l. Whe case ii) occurs, givig the first superset that ofte has the same colour as F, we iterate, brachig from this set to its supersets i the ext level. The operatio termiates whe case i) occurs, with all sets i the ext level havig the opposite colour. Brachig dow: Let A be a coloured set i F with d A, A 1 ) > ω, ad suppose d F, A i+1 ) > ω for all 1 i k 2 ad F A i. Brachig dow works exactly like brachig up, except 2 For a alterative etropic viewpoit, we could imagie that a adversary has chose a 2, k)-colourig of F, which we seek to determie by askig a series of questios. Our goal will the be to limit the amout of iformatio we receive, as we ca the boud the umber of 2, k)-colourigs the adversary had available to choose from. 14

15 we cosider subsets i the lower level, istead of supersets from the level above. Brachig up ad dow: Suppose some 2 i 0 k 2 has the property that d + F, A i 1 ) > ω for every 2 i i 0 ad F A i, ad d F, A i+1 ) > ω for every i i k 2 ad F A i. Let A be a coloured set with d + A, A i0 ) > ω ad d A, A i0 +1) > ω. We first brach up from A, passig through its supersets i A i0, ad coutig up through the higher parts. If we have ot ecoutered a moochromatic eighbourhood by the time we reach A 1, we brach dow from A, cosiderig its subsets i A i0 +1, ad the cotiuig through the lower parts. We ca also brach up ad dow from a coloured pair A B i A i0 +1 if d + B, A i0 ) > ω ad, as supposed, d A, A i0 +2) > ω. We the brach up through supersets of B i higher parts, ad the brach dow through subsets of A i lower parts. Note that with each brachig step i these operatios, wheever we ecouter case ii), we fid a loger moochromatic chai ivolvig the iitial set A. Thus these operatios must termiate withi k 1 brachig steps. The followig lemma quatifies the outcome of a brachig operatio. Lemma 3.6. If, whe brachig up, dow or up ad dow from a coloured set A or a pair A B, t further sets are added to R, the C shriks by a factor of at most 1 6 2εt. Proof. Assume we ecouter case ii) s times before ecouterig case i) ad edig the brachig operatio. As we argued previously, there ca be at most k 1 brachig steps, ad therefore we must have s k 2. Let j 1, j 2,..., j s be the idices of the moochromatic eighbours give i each istace of case ii). The umber of sets moved to R is the t j 1 + j j s + ω. Moreover, the set of 2, k)-colourigs C shriks by a factor of at most 1 2 ε ) ) 1 s 1 2 ε )2 εj i = 2 εj j 2 s) 1 2 ε ) s 1 2 εt ω) ε 2 ε 1 i=1 ) s+1 2 εt ω) 2 ε ε 2) s+1 < 2 εt ω+k) ε 2k = 2 εt ε 2 2 ε ) k 1 6 2εt, where the first iequality holds sice 2 x 1 + x 2 for every x [0, 1], the secod iequality sice s + 1 < k, ad the followig equality is due to the fact that ω = 4k log1/ε)/ε. Havig defied these brachig operatios, we are ow i a positio to specify how we obtai the partitio of Propositio 3.1. The procedure cosists for four stages. Stage I. The goal of this stage is to compress the 2k 2 atichais i 15) ito k 1 parts A 1,..., A k 1. These will o loger ecessarily be atichais, but we shall esure o set is cotaied i more tha ω other sets from its ow part. We do this greedily, shiftig a set up to a higher part wheever possible, ad the usig brachig operatios whe eeded. Shiftig up: Ruig i from 2 to 2k 2, cosider the sets i A i i the reverse of the predetermied liear order. Whe the set F is beig cosidered, if d + F, A i 1 ) ω, move F up to A i 1. We repeatedly ru the shiftig up operatio util o set is moved. At this poit, every set is cotaied i more tha ω sets from the part directly above. As a cosequece, if A i is o-empty for some i k, the sets i A i are i may k-chais. We shall use this fact to colour ad remove may sets efficietly via the followig operatio. 15

16 Colourig ad brachig up: Let l = max{i : A i }. If l k, let F 1 A l be the first set i A l with respect to. Colour F 1 with whichever colour occurs most frequetly i C breakig ties arbitrarily), say red. Restrict C to those colourigs where F 1 is red ad remove F 1 to R. We the brach up from F 1 to its supersets i A l 1, ad owards through higher parts. After ruig through this operatio, the sets removed may leave space from some lower sets to be shifted up. Hece we repeat this sequece of operatios util i k A i is empty, which marks the ed of Stage I. At this poit, we are left with the partitio F = A 1... A k 1 R, with the parts A i havig the followig two qualities i additio to Q1) from before. Q2) If F A i, the d + F, A i ) ω. Q3) If F A i for some 2 i k 1, the d + F, A i 1 ) > ω. The followig lemma shows these attributes do ideed hold after Stage I, ad that we do ot restrict our set of colourigs C too greatly. Lemma 3.7. At the ed of Stage I, we have the partitio F = A 1... A k 1 R with the qualities Q1) Q3). Moreover, the set C of colourigs shriks by a factor of at most 2 ε R. Proof. We first prove by cotradictio that our algorithm preserves the mootoicity of Q1). Suppose we are at the step whe mootoicity is violated, ad A, B) A i A j is the pair with i < j ad A B. It must be that j = i + 1 ad A has bee moved from A j to A i i the previous step via a shiftig up operatio, which implies d + A, A i ) ω. However, sice A B, we must have d + B, A i ) ω as well, ad A B. This meas that B would have bee moved from A j to A i before A, a cotradictio. We proceed by showig that Q2) is maitaied at every step of Stage I. If this is ot true, let us look closer at the first time whe there exists F A i with d + F, A i ) > ω. Iitially, A j is a atichai for every j 1, so Q2) holds at the begiig of Stage I. Hece either F or a set F i N + F, A i ) has bee moved up from A i+1 to A i i the previous step. The former case ca ot happe because we oly shift up if d + F, A i ) ω. O the other had, by Q1), we could ot have had F A i ad F A i+1 for ay F F, ad so the latter case is ruled out as well. As Stage I stops oly whe we ca o loger apply the shiftig up operatio, our partitio satisfies Q3) at the ed of Stage I. Fially, cosider the colourig ad brachig up operatios that we perform. Whe we colour the first set F 1, we choose the colour it most frequetly receives i the colourigs of C, which causes C to shrik by a factor of at most two. By Lemma 3.6, if t more sets are added to R by this operatio, C shriks by a further factor of at most 1 6 2εt. Hece whe R grows by a total of t sets, C shriks by a factor of at most 2 εt, thus esurig that C is at most 2 ε R times smaller by the ed of Stage I. Stage II. The previous stage gave us good cotrol, through Q1) Q3), of the umber of sets a give set is cotaied i. The goal of this stage is to obtai similar cotrol over the umber of sets a give set cotais. More precisely, at the ed of Stage II, we shall have a partitio F = i [k 1] A i U i D i ) R that has the three followig characteristics i additio to Q1) Q3). Q4) If F A i, the d F, A i ) ω. Q5) If F A i for some 1 i k 2, the d F, A i+1 ) > ω. 16

17 Q6) If U U i for some 2 i k 1, the max {du, A i ), du, A i 1 )} 2ω. If D D i for some 1 i k 2, the max {dd, A i ), dd, A i+1 )} 2ω. Moreover, U 1 = D k 1 =. Stage II cosists of two substages, of which we ow specify the first. Stage IIa. Set U i = D i = for every i [k 1]. We process the sets i A oe-by-oe, startig with the sets i A k 1 ad workig up to the sets i A 1. Withi a part A i, we cosider the sets accordig to the order. Assume we are curretly cosiderig a set F A i. There are three possibilities. Case 1: d F, A i ) ω ad, if i k 2, d F, A i+1 ) > ω. Sice F has qualities Q4) ad Q5), we leave F i place ad proceed to the ext set. Case 2: i k 2, d F, A i ) ω ad d F, A i+1 ) ω. I this case, F has quality Q4), ot cotaiig may sets from its ow part. However, it fails Q5), as it also does t cotai may sets from the level below. Hece we remove it from A i, ad place it i the part D i istead, to idicate it is dow-sparse. We the proceed to the ext set. Case 3: d F, A i ) > ω. I the fial case, F cotais may sets from its ow part. It is therefore cotaied i several k-chais, which we ca use to efficietly colour ad remove sets. However, this ca cause some of the previously established qualities to be violated, ad thus we must restart the stage after the brachig operatio. We explai i more detail below. Colourig ad brachig up ad dow: Colour F with its most frequet colour i C, say red. Restrict C to those colourigs where F is red ad remove F to R. We the apply the brachig up ad dow operatio from F with i 0 = i 1. Note that for ay set F F we must have F F, ad therefore F is yet to be cosidered, ad is thus still i A. This, together with Q3), implies we ca brach up. O the other had, for a set F F, we have F F, ad would therefore already have cosidered F i Stage IIa. If F is still i A j for some i j k 2, it must have falle uder Case 1, ad thus cotais more tha ω sets i A j+1, thereby esurig that the brachig dow part of the operatio ca also be carried out. However, whe we remove sets i the brachig operatio, we could destroy the qualities Q3) ad Q5) of sets that we have already cosidered, as the up- ad dow-degrees could decrease, ad hece we must restore these. For 1 i k 1, we retur ay sets i D i to the part A i. Note that sets which were coloured ad removed to R are ot retured. With D oce agai empty, we shift sets up to restore the quality Q3), just as we did i Stage I. Shiftig up: Startig from i = 2 ad ruig through to i = k 1, cosider the sets i A i i the reverse of the order. Whe dealig with F A i, if d + F, A i 1 ) ω, we move F to the part A i 1. Repeat this procedure util o sets are moved. After the shiftig up procedure termiates, we have agai restored Q1) Q3). At this poit, we ca restart Stage II, aimig to esure Q4) ad Q5) are satisfied as well. 17

18 Hece, to ed Case 3, we retur to the first set uder i A k 1 ad restart Stage IIa. Stage IIa eds oce we have goe through all the sets i A without ecouterig Case 3 which would cause us to restart the stage), after which we proceed to the secod part of Stage II. Observe that R grows mootoically with each iteratio, ad hece we ca oly restart Stage IIa a fiite umber of times before movig o to Stage IIb. Stage IIb. The partitio F = i [k 1] A i D i ) R at the ed of Stage IIa may o loger have Q3), as supersets of sets could have bee removed from A. To fix this issue, we move ay sets i A that violate either Q3) or Q5) to U or D respectively. Movig to U D: We process the sets i A accordig to the order. Give F A i, if i 2 ad d + F, A i 1 ) ω, move F to U i. If i k 2 ad d F, A i+1 ) ω, move F to D i. Repeat this process util o further sets are moved. We repeatedly apply this operatio util o sets are moved, which marks the ed of Stage IIb ad, with it, Stage II. Lemma 3.9 sumarises the effects of Stage II o our partitio, but we first itroduce some termiology that we will use throughout the remaider of this sectio. Defiitio 3.8. Give i [k 1] ad a subfamily B i of U i D i P i, the right-to-left order of the members of A i B i is obtaied by placig from right to left the sets of B i i the order i which they etered B i, followed by elemets of A i accordig to the order. Moreover, cosider the momet some set F is last 3 ) moved to U i D i P i. The prospective left-eighbourhood of F, deoted by N pr F ), is defied to be NF, A i A i 1 ) if F U i, NF, A i A i+1 ) if F D i, ad NF, A i ) if F P i, where the parts A i 1, A i ad A i+1 are take as they were durig the step whe F was removed from A. Eve though these parts could shrik durig subsequet steps of the process, we will ot update N pr F ) correspodigly. Lemma 3.9. At the ed of Stage II, the followig statemets are true. i) The partitio F = i [k 1] A i U i D i ) R has the qualities Q1) Q6). ii) For every F U D, N pr F ) 3ω. Moreover, suppose B i {U i D i, U i D i 1, D i U i+1 }. If F B i, the the left-eighbourhood of F with respect to the right-to-left order of the members of A i B i is a subfamily of N pr F ). iii) If t sets are coloured ad removed durig this stage, the family C of colourigs shriks by a factor of at most 2 εt. Proof. For i), we verify the various qualities i tur, startig with Q1) ad Q2). At the begiig of Stage II, the partitio satisfies these qualities, which ca oly be violated if sets are moved ito parts A i. This oly occurs durig Case 3 of Stage IIa, whe shiftig up ad whe emptyig D before restartig the stage. As proved i Lemma 3.7, the shiftig up procedure preserves Q1) ad Q2). Furthermore, i Stage IIa, whe D =, o sets are moved withi or ito A. Hece whe the sets from D are retured to their origial parts i A, they still have Q1) ad Q2). This shows that these qualities hold at the ed of Stage II. As Stage IIb oly termiates whe Q3) ad Q5) are satisfied for every set i A, it is evidet that these qualities hold. Q4) holds for every set i A at the ed of Stage IIa, sice otherwise we would have falle i Case 3 ad restarted the stage. As sets ca oly be removed 3 Note that P i = throughout Stage II. Furthermore, the oly way a set ca leave U D P is through Case 3 of Stage IIa. That apart, if a set is moved ito oe of these parts, it will remai there. 18