Tur\'an's Theorem for the Fano plane

TL;DR

This paper completely solves Turán's hypergraph problem for the Fano plane, identifying the unique extremal hypergraph structures for all sufficiently large n and describing the configurations for smaller n.

Contribution

It provides a full characterization of the extremal hypergraphs avoiding the Fano plane, confirming a conjecture and detailing the unique extremal configurations for all n ≥ 7.

Findings

For n ≥ 8, the extremal hypergraph is the balanced bipartite hypergraph.

For n=7, there are two extremal configurations: the bipartite hypergraph and a clique with five edges removed.

The result confirms the conjecture of Sós and extends previous partial results.

Abstract

Confirming a conjecture of Vera T. S\'os in a very strong sense, we give a complete solution to Tur\'an's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ 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 F\"uredi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.