Forbidding rank-preserving copies of a poset
Dániel Gerbner, Abhishek Methuku, Dániel T. Nagy, Balázs Patkós, Máté Vizer
aa r X i v : . [ m a t h . C O ] O c t Forbidding rank-preserving copies of a poset
D´aniel Gerbner a, ∗ , Abhishek Methuku b , D´aniel T. Nagy a, † ,Bal´azs Patk´os a, ‡ , M´at´e Vizer a, § a Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of SciencesP.O.B. 127, Budapest H-1364, Hungary. b Central European University, Department of MathematicsBudapest, H-1051, N´ador utca 9. { gerbner,nagydani,patkos } @renyi.hu, { abhishekmethuku,vizermate } @gmail.com September 28, 2018
Abstract
The maximum size, La ( n, P ), of a family of subsets of [ n ] = { , , ..., n } without con-taining a copy of P as a subposet, has been intensively studied.Let P be a graded poset. We say that a family F of subsets of [ n ] = { , , ..., n } containsa rank-preserving copy of P if it contains a copy of P such that elements of P having thesame rank are mapped to sets of same size in F . The largest size of a family of subsets of[ n ] = { , , ..., n } without containing a rank-preserving copy of P as a subposet is denotedby La rp ( n, P ). Clearly, La ( n, P ) ≤ La rp ( n, P ) holds.In this paper we prove asymptotically optimal upper bounds on La rp ( n, P ) for treeposets of height 2 and monotone tree posets of height 3, strengthening a result of Bukh inthese cases. We also obtain the exact value of La rp ( n, { Y h,s , Y ′ h,s } ) and La ( n, { Y h,s , Y ′ h,s } ),where Y h,s denotes the poset on h + s elements x , . . . , x h , y , . . . , y s with x < · · · < x h In extremal set theory, many of the problems considered can be phrased in the following way:what is the size of the largest family of sets that satisfy a certain property. The very first suchresult is due to Sperner [15] which states that if F is a family of subsets of [ n ] = { , . . . , n } (we write F ⊆ [ n ] to denote this fact) such that no pair F, F ′ ∈ F of sets are in inclusion F ( F ′ , then F can contain at most (cid:0) n ⌊ n/ ⌋ (cid:1) sets. This is sharp as shown by (cid:0) [ n ] ⌊ n/ ⌋ (cid:1) (the familyof all k -element subsets of a set X is denoted by (cid:0) Xk (cid:1) and is called the k th layer of X ). If P isa poset, we denote by ≤ P the partial order acting on the elements of P . Generalizing Sperner’sresult, Katona and Tarj´an [8] introduced the problem of determining the size of the largest family F ⊆ [ n ] that does not contain sets satisfying some inclusion patterns. Formally, if P is a finiteposet, then a subfamily G ⊆ F is • a (weak) copy of P if there exists a bijection φ : P → G such that we have φ ( x ) ⊆ φ ( y )whenever x ≤ P y holds, • a strong or induced copy of P if there exists a bijection φ : P → G such that we have φ ( x ) ⊆ φ ( y ) if and only if x ≤ P y holds.A family is said to be P -free if it does not contain any (weak) copy of P and induced P -free if it does not contain any induced copy of P . Katona and Tarj´an started the investigation ofdetermining La ( n, P ) := max {|F | : F ⊆ [ n ] , F is P -free } and La ∗ ( n, P ) := max {|F | : F ⊆ [ n ] , F is induced P -free } . The above quantities have been determined precisely or asymptotically for many classes of posets(see [6] for a nice survey), but the question has not been settled in general. Recently, Methukuand P´alv¨olgyi [11] showed that for any poset P , there exists a constant C P such that La ( n, P ) ≤ La ∗ ( n, P ) ≤ C P (cid:0) n ⌊ n/ ⌋ (cid:1) holds (the inequality La ( n, P ) ≤ | P | (cid:0) n ⌊ n/ ⌋ (cid:1) follows trivially from a resultof Erd˝os [4]). However, it is still unknown whether the limits π ( P ) = lim n →∞ La ( n,P ) ( n ⌊ n/ ⌋ ) and π ∗ ( P ) =lim n →∞ La ∗ ( n,P ) ( n ⌊ n/ ⌋ ) exist. In all known cases, the asymptotics of La ( n, P ) and La ∗ ( n, P ) were givenby “taking as many middle layers as possible without creating an (induced) copy of P ”. Thereforeresearchers of the area believe the following conjecture that appeared first in print in [7]. Conjecture 1.1. (i) For any poset P let e ( P ) denote the largest integer k such that for any j and n the family ∪ ki =1 (cid:0) [ n ] j + i (cid:1) is P -free. Then π ( P ) exists and is equal to e ( P ) .(ii) For any poset P let e ∗ ( P ) denote the largest integer k such that for any j and n the family ∪ ki =1 (cid:0) [ n ] j + i (cid:1) is induced P -free. Then π ∗ ( P ) exists and is equal to e ∗ ( P ) . P be a graded poset with rank function ρ . Given a family F , a subfamily G ⊆ F is arank-preserving copy of P if G is a (weak) copy of P such that elements having the same rankin P are mapped to sets of same size in G . More formally, G ⊆ F is a rank-preserving copy of P if there is a bijection φ : P → G such that | φ ( x ) | = | φ ( y ) | whenever ρ ( x ) = ρ ( y ) and we have φ ( x ) ⊆ φ ( y ) whenever x ≤ P y holds. A family F is rank-preserving P -free if it does not containa rank-preserving copy of P . In this paper, we study the function La rp ( n, P ) := max {|F | : F ⊆ [ n ] , F is rank-preserving P -free } . In fact, our problem is a natural special case of the following general problem introduced byNagy [13]. Let c : P → [ k ] be a coloring of the poset P such that for any x ∈ [ k ] the pre-image c − ( x ) is an antichain. A subfamily G ⊆ F is called a c -colored copy of P in F if G is a (weak)copy of P and sets corresponding to elements of P of the same color have the same size. Nagyinvestigated the size of the largest family F ⊆ [ n ] which does not contain a c -colored copy of P ,for several posets P and colorings c . Note that when c is the rank function of P , then this is equalto La rp ( n, P ). Nagy also showed that there is a constant C P such that La rp ( n, P ) ≤ C P (cid:0) n ⌊ n/ ⌋ (cid:1) . A complete multi-level poset is a poset in which every element of a level is related to everyelement of another level. Note that any rank-preserving copy of a complete multi-level poset P is also an induced copy of P . In fact, in [14], Patk´os determined the asymptotics of La ∗ ( n, P ),for some complete multi-level posets P by finding a rank preserving copy of P .By definition, for every graded poset P we have La ( n, P ) ≤ La rp ( n, P ). Boehnlein and Jiang[1] gave a family of posets P showing that the difference between La ∗ ( n, P ) and La ( n, P ) canbe arbitrarily large. Since their posets embed into a complete multi-level poset of height 3 in arank-preserving manner, the above mentioned result of Patk´os implies that for the same familyof posets, La rp ( n, P ) can be arbitrarily smaller than La ∗ ( n, P ). However, it would be interestingto determine if the opposite phenomenon can occur. Asymptotic results For a poset P its Hasse diagram , denoted by H ( P ), is a graph whose vertices are elements of P , and xy is an edge if x < y and there is no other element z of P with x < z < y . We call aposet, tree poset if H ( P ) is a tree. A tree poset is called monotone increasing if it has a uniqueminimal element and it is called monotone decreasing if it has a unique maximal element. A treeposet is monotone if it is either monotone increasing or decreasing.A remarkable result concerning Conjecture 1.1 is that of Bukh [2], who verified Conjecture1.1 (i) for tree posets. In the following results we strengthen his result in two cases. Theorem 1.2. Let T be any tree poset of height . Then we have La rp ( n, T ) = O T r log nn !! (cid:18) n ⌊ n ⌋ (cid:19) . heorem 1.3. Let T be any monotone tree poset of height 3. Then we have La rp ( n, T ) = O T r log nn !! (cid:18) n ⌊ n ⌋ (cid:19) . The lower bounds in Theorem 1.2 and Theorem 1.3 follow simply by taking one and twomiddle layers of the Boolean lattice of order n , respectively. An exact result The dual of a poset P is the poset P ′ on the same set with the partial order relation of P replacedby its inverse, i.e., x ≤ y holds in P if and only if y ≤ x holds in P ′ . Let Y h,s denote the poset on h + s elements x , . . . , x h , y , . . . , y s with x < · · · < x h < y , . . . , y s and let Y ′ h,s denote the dualof Y h,s . Let Σ( n, h ) for the number of elements on the h middle layers of the Boolean lattice oforder n , so Σ( n, h ) = P hi =1 (cid:0) n ⌊ n − h ⌋ + i (cid:1) .Investigation on La ( n, Y h,s ) was started by Thanh in [16], where asymptotic results wereobtained. Thanh also gave a construction showing that La ( n, Y h,s ) > Σ( n, h ), from which iteasily follows that La ( n, Y ′ h,s ) > Σ( n, h ) as well. Interestingly, De Bonis and Katona [3] showedthat if both Y , and Y ′ , are forbidden, then an exact result can be obtained: La ( n, { Y , , Y ′ , } ) =Σ( n, La ( n, { Y k, , Y ′ k, } ) =Σ( n, k ), and La ∗ ( n, { Y , , Y ′ , } ) = Σ( n, La ∗ ( n, { Y k, , Y ′ k, } ) = Σ( n, k ). Weprove the following theorem which extends all of these previous results and proves a conjectureof [10]. Theorem 1.4. For any pair s, h ≥ of positive integers, there exists n = n ( h, s ) such that forany n ≥ n we have La rp ( n, { Y h,s , Y ′ h,s } ) = Σ( n, h ) . The lower bound trivially follows by taking h middle layers of the Boolean lattice of order n .(Note that adding any extra set creates a rank-preserving copy of either Y h,s or Y ′ h,s .) Moreover,any rank-preserving copy of Y h,s (respectively Y ′ h,s ) is also an induced copy of Y h,s (respectively Y ′ h,s ). Therefore, Theorem 1.4 implies that La ∗ ( n, { Y h,s , Y ′ h,s } ) = La ( n, { Y h,s , Y ′ h,s } ) = Σ( n, h ). Remark. One wonders if the condition h ≥ La ( n, { Y , , Y ′ , } ) = (cid:0) nn/ (cid:1) if n is even and La ( n, { Y , , Y ′ , } ) = 2 (cid:0) n − n − / (cid:1) > (cid:0) nn/ (cid:1) if n is odd. The following construction shows that no matter how little we weaken the conditionof being { Y , , Y ′ , } -free, there are families strictly larger than (cid:0) nn/ (cid:1) even in the case n is even.Let us define F , = (cid:26) F ∈ (cid:18) [ n ] n/ (cid:19) : n − , n ∈ F (cid:27) ∪ (cid:26) F ∈ (cid:18) [ n ] n/ (cid:19) : | F ∩ { n − , n }| ≤ (cid:27) . Observe that F , is { Y , , Y ′ , } -free and its size is (cid:0) n − n/ (cid:1) + ( (cid:0) nn/ (cid:1) − (cid:0) n − n/ − (cid:1) ) > (cid:0) nn/ (cid:1) .4 Proofs Using Chernoff’s inequality, it is easy to show (see for example [7]) that the number of sets F ⊂ [ n ] of size more than n/ √ n log n or smaller than n/ − √ n log n is at most O (cid:18) n / (cid:18) nn/ (cid:19)(cid:19) . (1)Thus in order to prove Theorem 1.2 and Theorem 1.3, we can assume the family only containssets of size more than n/ − √ n log n and smaller than n/ √ n log n . The proof of Theorem 1.2 follows the lines of a reasoning of Bukh’s [2]. The new idea is that wecount the number of related pairs between two fixed levels as detailed in the proof below.Let F be a T -free family of subsets of [ n ] and let the number of elements in T be t . Using(1), we can assume F only contains sets of sizes in the range [ n/ − √ n log n, n/ √ n log n ].A pair of sets A, B ∈ F with A ⊂ B is called a 2-chain in F . It is known by a result of Kleitman[9] that the number of 2-chains in F is at least (cid:18) |F | − (cid:18) n ⌊ n ⌋ (cid:19)(cid:19) n . (2)For any n/ − √ n log n ≤ i ≤ n/ √ n log n , let F i := F ∩ (cid:0) [ n ] i (cid:1) . Claim 2.1. For any i < j , the number of 2-chains A ⊂ B with A ∈ F i and B ∈ F j is at most( t − |F i | + |F j | ). Proof. Suppose otherwise, and construct an auxiliary graph G whose vertices are elements of F i and F j , and two vertices form an edge of G if the corresponding elements form a 2-chain. Thisimplies that G contains more than ( t − |F i | + |F j | ) edges, so it has average degree more than2( t − G ′ of G with minimum degree at least t − 1, into whichwe can greedily embed any tree with t vertices. So in particular, we can find T in G ′ whichcorresponds to a rank-preserving copy of T into F , a contradiction.Claim 2.1 implies that the total number of 2-chains in F is at most X n/ − √ n log n ≤ i F ⊂ [ n ] be a family not containing a rank-preserving copy of Y h,s or Y ′ h,s . First, we willintroduce a weight function. For every F ∈ F , let w ( F ) = (cid:0) n | F | (cid:1) . For a maximal chain C , let w ( C ) = P F ∈C∩F w ( F ) denote the weight of C . Let C n denote the set of maximal chains in [ n ].Then 1 n ! X C∈ C n w ( C ) = 1 n ! X C∈ C n X F ∈C∩F w ( F ) = 1 n ! X F ∈F | F | !( n − | F | )! w ( F ) = |F | . This means that the average of the weight of the full chains equals the size of F . Thereforeit is enough to find an upper bound on this average. We will partition C n into some parts andshow that the average weight of the chains is at most Σ( n, h ) in each of the parts. Thereforethis average is also at most Σ( n, h ), when calculated over all maximal chains, which gives us |F | ≤ Σ( n, h ).Let G = { F ∈ F | ∃ P, Q ∈ F \{ F } , P ⊂ F ⊂ Q } . Let A ⊂ A ⊂ · · · ⊂ A h − be h − G . Then we define C ( A , A , . . . , A h − ) as the set of those chains that containall of A , A , . . . A h − and these are the h − G in them. We also define C − as the set of those chains that contain at most h − G . Then the sets of the form C ( A , A , . . . A h − ) together with C − are pairwise disjoint and their union is C n .Now we will show the average weight within each of these sets of chains is at most Σ( n, h ).This is easy to see for C − . If C ∈ C − , then |C ∩ F | ≤ h , since every element of F ∩ C except forthe smallest and the greatest must be in G . Therefore c ( W ) ≤ Σ( n, h ) for every C ∈ C − , whichtrivially implies X C∈ C − w ( C ) ≤ | C − | Σ( n, h ) . Now consider some sets A ⊂ A ⊂ · · · ⊂ A h − in G such that C ( A , A , . . . A h − ) is non-empty. We will use the notations C ( A , A , . . . , A h − ) = Q , | A | = ℓ and n − | A h − | = ℓ forsimplicity. Note that the chains in Q do not contain any member of F of size between | A | and | A h − | other than the sets A , A . . . A h − . Such a set would be in G (since it contains A and iscontained in A h − ), therefore its existence would contradict the minimality of { A , A , . . . , A h − } .The chains in Q must also avoid all subsets of A that are in G for the same reason.Let N denote the number of chains between ∅ and A that avoid the elements of G (ex-cept for A ). Let N denote the number of chains between A and A h − that contain the sets A , A , . . . , A h − , but no other element of F . Then |Q| = N N ℓ !.Now we will investigate how much the sets of certain sizes can contribute to the sum X C∈Q w ( C ) . (5)8he sets A , A , . . . A h − appear in all chains of Q , so their contribution to the sum is |Q| h − X i =1 w ( A i ) = |Q| h − X i =1 (cid:18) n | A i | (cid:19) ≤ |Q| Σ( n, h − . We have already seen that there are no other sets of F in these chains with a size between | A | and | A h − | .If ℓ < n − √ n log n , then (by (1)) the contribution coming from the subsets of A is triviallyat most |Q| ℓ − X i =0 (cid:18) ni (cid:19) = |Q| O (cid:18)(cid:18) nn/ (cid:19) n / (cid:19) . The contribution coming from supersets of A h − is similarly small if ℓ < n − √ n log n . Fromnow on we consider the cases when ℓ ≥ n − √ n log n and ℓ ≥ n − √ n log n .There are no s supersets of A h − of equal size in F , since these would form a rank-preservingcopy of Y h,s together with the sets A , A , . . . A h − and some set P ∈ F , P ⊂ A . (Such a setexists, since A ∈ G .)A superset of A h − of size n − i appears in |Q| (cid:0) ℓ i (cid:1) − chains of Q . Therefore the totalcontribution to the sum (5) by supersets of A h − is at most |Q| w ([ n ]) + ℓ − X i =1 |Q| (cid:18) ℓ i (cid:19) − ( s − (cid:18) nn − i (cid:19) ≤ |Q| + |Q| ( s − (cid:18) n ⌊ n ⌋ (cid:19) ℓ − X i =1 (cid:18) ℓ i (cid:19) − = |Q| (cid:18) n ⌊ n ⌋ (cid:19) O s (cid:18) n (cid:19) . There are no s subsets of A of equal size in F , since these would form a rank-preservingcopy of Y ′ h,s together with the sets A , A , . . . A h − and some set Q ∈ F , A h − ⊂ Q . (Such a setexists, since A h − ∈ G .)A subset of A of size i appears in at most (cid:0) ℓ i (cid:1) − ℓ ! N ℓ ! chains of Q . Therefore the totalcontribution to the sum (5) by subsets of A is at most ℓ ! N ℓ ! w ( ∅ ) + ℓ − X i =1 (cid:18) ℓ i (cid:19) − ℓ ! N ℓ !( s − (cid:18) ni (cid:19) ≤ ℓ ! N ℓ ! + ℓ ! N ℓ !( s − (cid:18) n ⌊ n ⌋ (cid:19) ℓ − X i =1 (cid:18) ℓ i (cid:19) − = ℓ ! N ℓ ! (cid:18) n ⌊ n ⌋ (cid:19) O s (cid:18) n (cid:19) . (6)We will show that if n is large and ℓ ≥ n − √ n log n then most chains between ∅ and A avoid the elements of G , therefore N is close to ℓ !. There are at most s − G on any9evel (otherwise a rank-preserving copy of Y ′ h,s would be formed), and ∅ 6∈ G . There are ℓ ! (cid:0) ℓ i (cid:1) − chains between ∅ and A containing a set of size i . Therefore ℓ ! − N ≤ ( s − ℓ − X i =1 ℓ ! (cid:18) ℓ i (cid:19) − = ℓ ! O (cid:18) n (cid:19) . This means that for large enough n , we have ℓ ! ≤ N . Then (6) can be continued as ℓ ! N ℓ ! (cid:18) n ⌊ n ⌋ (cid:19) O s (cid:18) n (cid:19) ≤ N N ℓ ! (cid:18) n ⌊ n ⌋ (cid:19) O s (cid:18) n (cid:19) = |Q| (cid:18) n ⌊ n ⌋ (cid:19) O s (cid:18) n (cid:19) . To summarize, we found that the contribution to the sum (5) from the subsets of A and thesupersets of A h − is at most |Q| (cid:18) n ⌊ n ⌋ (cid:19) O s (cid:18) n (cid:19) . For large enough n this is smaller than |Q| (Σ( n, h ) − Σ( n, h − X C∈Q w ( C ) ≤ |Q| Σ( n, h ) . This completes the proof. Remark. We had to use a weighting technique in the above proof because the usual Lubellmethod (proving that P F ∈F (cid:0) n | F | (cid:1) − ≤ h , and deducing |F | ≤ Σ( n, h ) from that) does not workfor this problem. To see this, let h ≥ n ≥ h and consider the following set system: F = { F ∈ [ n ] | | F | ≤ h − | F | ≥ n − h + 2 } . For s ≥ h − this set system is Y h,s -free and Y ′ h,s -free (even in the original sense, not necessarilyin the rank-preserving sense). However, we have P F ∈F (cid:0) n | F | (cid:1) − = 2( h − > h . References [1] Boehnlein, E., and Jiang, T. 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