The codegree threshold of K-4

• The codegree threshold ex2 (n, F) of a 3-graph F is the minimum d = d(n) such that every 3-graph on n vertices in which every pair of vertices is contained in at least d + 1 edges contains a copy of F as a subgraph. We study ex2 (n, F) when F = K-4 , the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that if n is sufficiently large, thenex2 (n, K-4)⩽ (n + 1)/4.This settles in the affirmative a conjecture of Nagle [Congressus Numerantium, 1999, pp. 119-128]. In addition, we obtain a stability result: for every near-extremal configuration G, there is a quasirandom tournament T on the same vertex set such that G is o(n3)-close in the edit distance to the 3-graph C(T) whose edges are the cyclically oriented triangles from T. For infinitely many values of n, we are further able to determine ex2(n, K-4) exactly and to show that tournament-based constructions C(T) are extremal for those values of n.

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NATURVETENSKAP  -- Matematik -- Diskret matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Discrete Mathematics (hsv//eng)

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