Lecture notes for Discrete mathematics of Paul Erd}os- NDMI107 Copyright c VaˇsekChv´atal2020 EXTREMAL SET THEORY 1 Sperner's theorem The Erd}os-Radoinvestigations of ∆-systems constitute a part of extremal set theory, which concerns extremal sizes of set families with prescribed properties (such as including only k-point sets and containing no ∆-system of more than m sets). A classic of this theory comes from Emanuel Sperner (1905 { 1980): Theorem 1.1 (Sperner [49]). Let n be a positive integer. If V is an n-point set and E is a family of subsets of V such that S; T 2 E;S 6= T ) S 6⊆ T; (1) then n jEj ≤ : (2) bn=2c Furthermore, (2) holds as equality if and only if E consists of all subsets of V that have size bn=2c or all subsets of V that have size dn=2e. A family E of sets with property (1) is called an antichain. The original proof of Theorem 1.1 involves an intermediate result of independent interest: Lemma 1.1. Let n and k be integers such that 1 < k < n; let V be an n-point set, let F be a nonempty family of k-point subsets of V , and let F (+), F (−) be defined by F (+) = fT ⊆ V : jT j = k + 1 and S ⊂ T for some S in Fg; F (−) = fR ⊆ V : jRj = k − 1 and R ⊂ S for some S in Fg: Then (i) if k < (n − 1)=2, then jF (+)j > jFj, n (+) (ii) if k = (n − 1)=2 and jFj < k , then jF j > jFj, (iii) if k > (n + 1)=2, then jF (−)j > jFj, n (−) (iv) if k = (n + 1)=2 and jFj < k , then jF j > jFj. Proof. Let N denote the number of all pairs (S; T ) such that S 2 F, S ⊂ T ⊆ V , and jT j = k + 1. Each T in F (+) appears in at most k + 1 such pairs, and so 1 N ≤ jF (+)j · (k + 1); each S in F appears in precisely n − k such pairs, and so N = jFj · (n − k); we conclude that jF (+)j N n − k ≥ ≥ : (3) jFj (k + 1)jFj k + 1 Proof of (i): Under the assumption k < (n − 1)=2, we have (n − k)=(k + 1) > 1 and the conclusion follows from (3). Proof of (ii): Here, (3) is transformed into jF (+)j N > = 1 : jFj (k + 1)jFj the assumption k = (n − 1)=2 means that (n − k)=(k + 1) = 1 and under the n (+) assumption jFj < k , we will prove that N < jF j · (k + 1). To do this, we will find a T in F (+) with at least one k-point subset outside F. Since jFj > 0, n there is a k-point subset S of V such that S 2 F; since jFj < k , there is a k-point subset S0 of V such that S0 62 F; replacing points in S − S0 by points 0 in S − S one by one, we construct a sequence S0;S1;:::;St of k-point subsets 0 Si of V such that S0 = S, St = S , and jSi−1 \ Sij = k − 1 for all i = 1; 2; : : : ; t. If i is the smallest subscript such that Si 62 F, then Si−1 2 F and we can take T = Si [ Si−1. Proofs of (iii) and (iv): In (i) and (ii), replace F by fV − S : S 2 Fg. Proof of Theorem 1.1. Let n be a positive integer, let V be an n-point set, and let E be any largest antichain of subsets of V ; write kmin = minfjSj : S 2 Eg; kmax = maxfjSj : S 2 Eg and Emin = fS 2 E : jSj = kming; Emax = fS 2 E : jSj = kmaxg: (+) (+) Since E is an antichain, Emin is disjoint from E and (E − Emin) [Emin is an (+) antichain; since E is a largest antichain of subsets of V , it follows that jEminj ≤ jEminj; now part (i) of Lemma 1.1 guarantees that (i) kmin ≥ (n − 1)=2 and part (ii) of Lemma 1.1 guarantees that n (ii) if kmin = (n − 1)=2, then jEminj = (n−1)=2 . Similar arguments show that (iii) kmax ≤ (n + 1)=2 and that n (iv) if kmax = (n + 1)=2, then jEmaxj = (n+1)=2 . The conclusion of the theorem follows from (i), (ii), (iii), (iv). 2 1.1 A simple proof of Sperner's theorem Independently of each other, Koichi Yamamoto [55], Lev Dmitrievich Meshalkin (1934{2000) [38], and David Lubell [37] found the following inequality: Theorem 1.2 (The LYM inequality). If E is an antichain of subsets of an n-point set, then X 1 n ≤ 1: (4) S2E jSj Proof. Let V be an arbitrary but fixed n-point set and let E be an arbitrary but fixed antichain of subsets of V .A chain of length n + 1 is any family S0;S1;:::;Sn of subsets of V such that jSij = i for all i and S0 ⊂ S1 ⊂ ::: ⊂ Sn (in particular, S0 = ; and Sn = V ). We will count in two different ways the number N of all pairs (C;S) such that C is a chain of length n+1 and S 2 E \C. For each k-point subset S of V , there are precisely k! choices of a sequence S0, S1,. , Sk of subsets of V such that jSij = i for all i and S0 ⊂ S1 ⊂ ::: ⊂ Sk = S; there are precisely (n − k)! choices of a sequence Sk;Sk+1;:::;Sn of subsets of V such that jSij = i for all i and S = Sk ⊂ Sk+1 ⊂ ::: ⊂ Sn; it follows that each member S of our E participates in precisely jSj!(n − jSj)! of our pairs (C;S), and so X N = jSj!(n − jSj)! : S2E Since each chain contains at most one member of any antichain, each C partici- pates in at most one of our pairs (C;S), and so N is at most the number of all chains of length n + 1: N ≤ n! : Comparing the exact formula for N with this upper bound, we get the inequality X jSj!(n − jSj)! ≤ n! ; S2E which is just another way of writing (4). Let us prove Theorem 1.1 along the lines of Theorem 1.2. The first part of Theorem 1.1 | inequality (2) | is a direct consequence of Theorem 1.2: every antichain E of subsets of an n-point set satisfies n X X bn=2c n X 1 n jEj = 1 ≤ = ≤ : (5) n bn=2c n bn=2c S2E S2E jSj S2E jSj To prove the second part of Theorem 1.1 | characterization of extremal an- tichains | along these lines, consider an arbitrary antichain E of subsets of an n-point set V that satisfies both inequalities in (5) with the sign of equality. n n The first of these equations implies that jSj = bn=2c for all S in E, which 3 means that jSj = n=2 when n is even and jSj = (n ± 1)=2 when n is odd. If n is even, then we are done; if n is odd, then we will argue about the second inequality-turned-equation in (5). This equality means equality in (4); review- ing the proof of Theorem 1.2, we find that • every chain of length n + 1 contains a member of E, and so (since all sets in E have size (n ± 1)=2) • for every two subsets S; T of V such that jSj = (n − 1)=2, jT j = (n + 1)=2, and S ⊂ T , precisely one of S and T belongs to E. It follows at once that • jSj = jS0j = (n − 1)=2, S 2 E, S0 ⊂ V , jS [ S0j = (n + 1)=2 ) S0 2 E (consider T = S [ S0) and then induction on jS [ S0j shows that • jSj = jS0j = (n − 1)=2, S 2 E, S0 ⊂ V ) S0 2 E. 2 The Erd}os-Ko-Rado theorem In 1938, Paul Erd}os, Chao Ko, whose name is also transliterated as Ke Zhao (1910{2002), and Richard Rado proved a theorem that they published twenty- three years later [19, Theorem 1]. Its simplified version presented below is known as the Erd}os-Ko-Rado theorem. Theorem 2.1. Let n and k be positive integers such that 2k ≤ n. If V is an n-point set and E is a family of k-point subsets of V such that S; T 2 E ) S \ T 6= ;; (6) then n − 1 jEj ≤ : (7) k − 1 A family E of sets with property (6) is called an intersecting family. The original proof of Theorem 2.1 involves an intermediate result of independent interest: Lemma 2.1. Let V be a set and let E be an intersecting family of subsets of V. Given two elements x; y of V , write E∗ = fS 2 E : x 2 S; y 62 S; (S − fxg) [ fyg 62 Eg and define f : E! 2V by (S − fxg) [ fyg if S 2 E∗, f(S) = S otherwise. Then ff(S): S 2 Eg is an intersecting family of size jEj. 4 Proof. We will verify that S; T 2 E ) f(S) \ f(T ) 6= ; and that S 6= T ) f(S) 6= f(T ). Case 1: S; T 2 E − E∗. In this case, f(S) = S, f(T ) = T , and so both implications hold trivially. Case 2: S; T 2 E∗. In this case, we have f(S)\f(T ) 6= ; since y 2 f(S)\f(T ). In addition, S = (f(S) − fyg) [ fxg, T = (f(T ) − fyg) [ fxg, and so f(S) = f(T ) ) S = T .