A classical object in hypergraph Tur\'{ a}n theory is the Fano plane $\mathbb{ F}$, the unique linear $3$-graph on seven vertices with seven edges. The Tur\'{ a}n density and exact Tur\'{ a}n number of $\mathbb{ F}$, first proposed as a problem by S\'{ o}s \cite{ Sos76} in the 1970s, were determined through a sequence of works by De Caen-F\"{ u}redi \cite{ DCF00}, F\"{ u}redi-Simonovits \cite{ FS05}, Keevash-Sudakov \cite{ KS05}, and Bellmann-Reiher \cite{ BR19}. Addressing a conjecture of Balogh-Clemen-Lidick\'{ y} \cite[Conjecture 3.1]{ BCL22a}, we establish an Andr\'{ a}sfai-Erd\H{ o}s-S\'{ o}s-type stability theorem for $\mathbb{ F}$ in the $\ell_2$-norm: there exists a positive constant $\varepsilon$ such that for large $n$, every $\mathbb{ F}$-free $3$-graph on $n$ vertices with minimum $\ell_2$-norm degree at least $(5/4 - \varepsilon)n^3$ 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 Tur\'{ a}n problem of $\mathbb{ F}$, thereby confirming the conjecture of Balogh-Clemen-Lidick\'{ y}. Our proof includes a refinement of a classical result by Ahlswede-Katona \cite{ AK78} on counting stars, and the establishment of an Andr\'{ a}sfai-Erd\H{ o}s-S\'{ o}s-type theorem for a multigraph Tur\'{ a}n problem studied by Bellmann-Reiher \cite{ BR19}, both of which are of independent interest.
Categories: math.CO