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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)
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)
We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every n-vertex graph admits a separating path system of size O(n) and prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs of all densities and graphs with linear minimum(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)
In recent breakthrough results, Saxton–Thomason and Balogh–Morris–Samotij have developed powerful theories of hypergraph containers. These theories have led to a large number of new results on transference, and on counting and characterising typical graphs in hereditary properties. In a different direction, Hatami–Janson–Szegedy proved results on the(More)