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

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}. Addressing a conjecture of Balogh-Clemen-Lidický \cite[Conjecture 3.1]{BCL22a}, we establish an Andrásfai-Erdős-Sós-type stability theorem for $\mathbb{F}$ in the ${\mathrm{\ell }}_2$-norm: there exists a positive constant $\epsilon$ such that for large $n$, every $\mathbb{F}$-free $3$-graph on $n$ vertices with minimum ${\mathrm{\ell }}_2$-norm degree at least $(5\mathrm{/}4-\epsilon )n^3$ must be bipartite. As a consequence, for large $n$, the balanced complete bipartite $3$-graph is the unique extremal construction for the ${\mathrm{\ell }}_2$-norm Turán problem of $\mathbb{F}$, thereby confirming the conjecture of Balogh-Clemen-Lidický. 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.