A problem of Erdős and Sós on 3-graphs

Abstract

Abstract. We show that for every ε > 0 there exist δ > 0 and n0 ∈ N such that every 3-uniform hypergraph on n ≥ n0 vertices with the property that every k-vertex subset, where k ≥ δn, induces at least 1 4 + ε k 3 edges, contains K − 4 as a subgraph, where K − 4 is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sós. The constant 1/4 is the best possible.

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

“…Flag algebras can be used as a general tool to attack problems from extremal combinatorics. Flag algebras were used for a wide range of problems, for example the Caccetta-Häggkvist conjecture [15,21], Turán-type problems in graphs [7,11,13,19,23,26,27], 3-graphs [9,10] and hypercubes [1,3], extremal problems in a colored environment [2,4,6], and also to problems in geometry [17] or extremal theory of permutations [5]. For more details on these applications, see a recent survey of Razborov [24].…”

The codegree threshold ofK4−

“…Füredi observed however that the tournament construction of Erdős and Hajnal described above gives a negative answer to this question: a linear-density of at least 1/4 is required for the existence of a K − 4 -subgraph. In recent work, Glebov, Král' and Volec [12] showed this 1/4 lower bound is tight, using flag algebraic techniques amongst other ingredients in their proof. It also follows that the Erdős-Hajnal construction is asymptotically the unique K − 4 -free 1/4-linear dense 3-graph.…”

Decomposing Graphs into Edges and Triangles

“…The flag algebra method introduced by Razborov [33] has changed the landscape of extremal combinatorics. It has been applied to many long-standing open problems, for example [1,2,3,4,5,6,8,9,10,12,13,14,15,16,21,23,27,28,29,30]. The method is designed to analyse asymptotic behaviour of substructure densities and we now briefly describe it.…”