The Tur\'an number of $F_{3,3}$

TL;DR

This paper determines the maximum number of edges in an $F_{3,3}$-free 3-graph for all sufficiently large n, providing a precise Turán number that improves previous bounds.

Contribution

The authors exactly compute the Turán number for $F_{3,3}$-free 3-graphs, refining earlier asymptotic results and establishing a sharp bound for all n ≥ 6.

Findings

Exact Turán number for $F_{3,3}$-free 3-graphs for all n ≥ 6

The maximum edges are given by a specific binomial coefficient formula

The result sharpens previous asymptotic bounds by Zhou and others.

Abstract

Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all $n \ge 6$, the maximum number of edges in an $F_{3,3}$-free 3-graph on $n$ vertices is $\binom{n}{3} - \binom{\lfloor n/2 \rfloor}{3} - \binom{\lceil n/2 \rceil}{3}$. This sharpens results of Zhou and of the second author and R\"odl.