Turán’s Theorem for the Fano Plane

Abstract

Confirming a conjecture of Vera T. Sós in a very strong sense, we give a complete solution to Turán's hypergraph problem for the Fano plane. That is we prove for n ě 8 that among all 3-uniform hypergraphs on n vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for n " 7 there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order 7 by removing all five edges containing a fixed pair of vertices.For sufficiently large values n this was proved earlier by Füredi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.2010 Mathematics Subject Classification. 05C65, 05D05.

Hypergraph Turán Problems in $\ell_2$-Norm

“…Frankl and Füredi [31] proved that the 3-graph with the maximum number of edges only containing exactly 0 or 2 edges on any 4 vertices is the n-vertex blow-up of H 6 . As observed in Subsection 3.11, this 3-graph has codegree squared sum of 5/108n 4 where E 3 4 denotes the 3-graph with exactly one edge on four vertices.…”

“…This problem was solved asymptotically by Caen and Füredi [16]. Later, Füredi and Simonovits [36] and, independently, Keevash and Sudakov [47] determined the extremal hypergraph for large n. Recently, Bellmann and Reiher [4] solved the question for all n.…”

“…We propose to study σ(H) for k-uniform hypergraphs with k ≥ 4. In particular it would be interesting to determine the codegree squared density for K 4 5 , the 4-uniform hypergraph on five vertices with five edges. For the Turán density, Giraud [38] proved π(K 4 5 ) ≥ 11/16 using the following construction.…”

“…Here, we will present two examples where we can determine exco ind 2 (n, F). Denote by K 4− uniform hypergraph with exactly 4 edges on 5 vertices and E4 5 the 4-uniform hypergraph with exactly one edge on five vertices. Observe that the proof of Theorem 8.3 actually gives the stronger result exco ind 2 (n, {E 4 5 , K 4= 5 , K 4− 5 , K 4 5 }) = n 2 16 n 3…”

Extremal problems in uniformly dense hypergraphs

“…By combining their work with Simonovits's stability method [33] it was shown in [15,20] that the conjecture holds for all sufficiently large hypergraphs. A full proof applying to all n ě 7 was recently obtained in [4].…”