On the Exact Tur\'an Number of $F^-_{4,3}$

Abstract

For a $3$-graph $F$, the Tur\'an number of $F$, denoted by $\ex(n,F)$, is the maximum number of edges in a $3$-graph on $n$ vertices containing no subgraph isomorphic to $F$. Let $F^-_{4,3}$ be the $3$-graph formed by a complete four-vertex core and three outer vertices, with all but one of the twelve triples containing one core vertex and two outer vertices. We prove that, for every $n\ge8$, \[ \ex(n,F^-_{4,3})=\binom n3-\binom{\lfloor n/2\rfloor}{3}-\binom{\lceil n/2\rceil}{3}, \] and the balanced complete bipartite $3$-graph is the unique extremal configuration. This determines the exact value and all equality cases in the asymptotic theorem of Mubayi and R\"odl. It also extends the exact Tur\'an Number of $F_{3,3}$ and resolves a conjecture of Frankl, Huang and R\"odl.