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A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the tetrahedron exhibits this phenomenon assuming Turán's conjecture. Our main result is to construct a finite family of triple systems ℳ , determine its Turán number, and prove that there are two near-extremal ℳ -free constructions that are far from each other in edit-distance. This is the first extremal result for a hypergraph family that fails to have a corresponding stability theorem. [ABSTRACT FROM AUTHOR]

A conjecture of Chung and Graham states that every K4‐free graph on n vertices contains a vertex set of size ⌊n∕2⌋ that spans at most n2∕18 edges. We make the first step toward this conjecture by showing that it holds for all regular graphs. [ABSTRACT FROM AUTHOR]

Let 3 ≤ d ≤ k and ν ≥ 0 be fixed. Let F ⊂ [ n ] k . The matching number of F , denoted by ν (F) , is the maximum number of pairwise disjoint sets in F. F is d -cluster-free if it does not contain d sets with union of size at most 2 k and empty intersection. In this paper, we give a lower bound and an upper bound for the maximum size of a d -cluster-free family with a matching number at least ν + 1. In particular, our result of the case ν = 1 settles a conjecture of Mammoliti and Britz. We also introduce a Turán problem in hypergraphs that allows multiple edges, which may be of independent interest. [ABSTRACT FROM AUTHOR]