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Let Ω be a finite set and let S ⊆ P(Ω) be a set system on Ω. For x ∈ Ω, we denote by d S (x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then there is some x ∈ Ω with d S (x) ≥ 1 2 |S|. We consider a related question. Define the weight of a family S to be w(S) := A∈S |A|. Suppose S is… (More)

Given a family of 3-graphs F, we define its codegree threshold coex(n, F) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F 3,2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe and cde. In this… (More)

Given a 3-graph F , its codegree threshold co-ex(n, F) is the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d triples but which contains no member of F as a subgraph. The limit γ(F) = lim n→∞ co-ex(n, F) n − 2 is known to exist and is called the codegree density of F. In this… (More)

Let V be an n-set, and let X be a random variable taking values in the powerset of V. Suppose we are given a sequence of random coupons X 1 , X 2 ,. . ., where the X i are independent random variables with distribution given by X. The covering time T is the smallest integer t ≥ 0 such that t i=1 X i = V. The distribution of T is important in many… (More)

Let G = (V, E) be a graph of density p on n vertices. Following Erd˝ os, Luczak and Spencer [11], an m-vertex subgraph H of G is called full if H has minimum degree at least p(m − 1). Let f (G) denote the order of a largest full subgraph of G. If p n 2 is a positive integer, define

Given two 3-uniform hypergraphs F and G, we say that G has an F-covering if we can cover V (G) by copies of F. The minimum codegree of G is the largest integer d such that every pair of vertices from V (G) is contained in at least d triples from E(G). Define c 2 (n, F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no… (More)

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