The maximum number of disjoint pairs in a family of subsets

Let .~" be a family of 2 "+1 subsets of a 2n-element set. Then the number of disjoint pairs in ~" is bounded by (1 + o(1))22". This proves an old conjecture of Erd/Ss. Let ~" be a family of 2 tl/tk+l~+~" subsets of an n-element set. Then the number of containments in ,~ is bounded by / I~1 'X (1 1/k + o(1)) k 2 :" This verifies a conjecture of Daykin and Erd/Ss. A similar Erd6s-Stone type result is proved for the maximum number of disjoint pairs in a family of subsets. I. I n t r o d u c t i o n Let : be a family of m distinct subsets of X = {1, 2 . . . . . n}. Let d(,~) ( c ( : ) ) denote the number of disjoint (comparable, respectively) pairs in ~-. Tha t is: a(:) = I { { F , r } : F , F ' ~ , F n e ' = O~1 c(:) = I{(F,F'): F , F ' ~ , F = F'}I. Define d(n,m) = max{d(~ ) : I,~1 = m}, c(n,m) = max{c (~ ) : I~-I = m}. Several years ago Erd/Ss 14] raised the problems of determining or estimating d(n, m). A similar problem for c(n, m) is considered in i3]. Example 1.1. Let X 1 U X 2 U.o. U X k be a par t i t ion of X, where [n/k] < Igil < Fn/k'l. Suppose m < k '2 t'/kj and let ~r be a collection of subsets of Xi with [mlkJ < Idil < Fmlk], I~r + 1~/21 + " " + I~r = m. Then 1 m * Research supported in part by the Weizmann Fellowship for Scientific Research. 14 N. Alon and ]P. Frankl Example 1.2. Let X 1, X 2 . . . . . g k and m be as in Example 1.1. Let ~ be a collection of subsets of X 1UX 2 13...13X~, each containing X1 13X2 13...UXi-1, where [m/kJ < I~1 -< [m/k], Is~xl + 1821 + " " § I~l -m. Then ( 1 ) ( m ) 2" The following theorems show that these examples are essentially best possible. Theorem 1.3. For every positive integer k there exists a positive fl = fl(k) such that if m = 2 tl/tk+l)+6)n where 6 > 0 then 1 m d ( n , m , < ( 1 ~ ) ( 2 ) + 0(m2-a'2,. Theorem 1.4. For every positive integer k there exists a positive f f = if(k) such that i f m = 2 ~x/r where 6 > 0 then 1 m c ( n , m ) < ( 1 ~ ) ( 2 ) + 0(m2-a'n'*'). The case/~ = 1 of the above theorems was conjectured by Daykin and Erd6s [7]. The general case settles a problem of Erd6s [7 1 namely it shows that an ErdSsStone type result holds. Also the case k = 2 implies another conjecture of Erd6s in a much stronger form [4]. Our paper is organized as follows. In Section 2 we outline a direct probabilistic proof for the case k = 1 of both theorems. In Section 3 a partition lemma for arbitrary families of sets is proved and applied to verify the theorems with a somewhat weaker estimate of the remainder term. In Section 4 we combine the probabilistic approach with results of Bollob~s, Erd6s and Simonovits on supersaturated graphs and hypergraphs to prove both theorems in full strength. In Section 5 we outline various generalizations dealing with the number of chains of given length, the number of pairwise disjoint r-tuples of sets, etc. In the last section open problems and conjectures are mentioned. 2. The Basic Probabilistic Argument Let ~ be a family consisting of m = 2 ttl/2)+a)" subsets of X = {1,2 . . . . . n}, where 6 > O. We claim that d($ r) < m 2-62/2. (2.1) Note that this inequality, when applied to #" U {X F: F ~ a r } shows that C(#-) < 4m 2-62/2. To prove (2.1) suppose it is false and pick independently t members A 1, A 2 . . . . . At of #" with repetitions at random, where t is a large positive integer, to be chosen later. We will show that with positive probabilitylA 1 U A 2 13... 13 A,I > n/2 and still this union is disjoint to more than 2 n/2 distinct subsets of X. This contradiction will The Maximum Number of Disjoint Pairs in a Family of Subsets 15 establish (2.1). In fact