Exact Turán number of the Fano plane in the ${\ell }_2$-norm

Abstract

A classical object in hypergraph Turán theory is the Fano plane $\mathbb{F}$, the unique linear $3$-graph on seven vertices with seven edges. The Turán density and exact Turán number of $\mathbb{F}$, first proposed as a problem by Sós~\cite{Sos76} in the 1970s, were determined through a sequence of works by De Caen--Füredi~\cite{DCF00}, Füredi--Simonovits~\cite{FS05}, Keevash--Sudakov~\cite{KS05}, and Bellmann--Reiher~\cite{BR19}. Confirming a conjecture of Balogh--Clemen--Lidický~{\cite[Conjecture~3.1]{BCL22a}} in a stronger form, we establish an Andrásfai--Erdős--Sós--type stability theorem for $\mathbb{F}$ in the ${\ell }_2$-norm: there exists a positive constant $\epsilon$ such that for large $n$, every $\mathbb{F}$-free $3$-graph on $n$ vertices with minimum ${\ell }_2$-norm degree at least $(5/4-\epsilon )n^3$ must be bipartite. As a consequence, for large $n$, the balanced complete bipartite $3$-graph is the unique extremal construction for the ${\ell }_2$-norm Turán problem of $\mathbb{F}$. Our proof includes a refinement of a classical result by Ahlswede--Katona~\cite{AK78} on counting stars, and the establishment of an Andrásfai--Erdős--Sós--type theorem for a multigraph Turán problem studied by Bellmann--Reiher~\cite{BR19}, both of which are of independent interest.

arXiv comment37 pages, comments are welcome