The Brown-Erdős-Sós conjecture in dense triple systems.

Abstract: In 1973, Brown, Erdős, and Sós conjectured, in an equivalent form, that for any $\delta > 0$ and integer $e \geq 3$, every sufficiently large linear 3-uniform hypergraph with at least $\delta n^2$ edges contains a collection of $e$ edges whose union spans at least $e+3$ vertices. In this talk, we show a proof of the conjecture for the case $\delta > 4/5$.