A counterexample to the Erdős--Sós bipartite-link conjecture

Erdős and Sós conjectured that every $n$-vertex $3$-uniform hypergraph whose link graphs are all bipartite has at most $(1\mathrm{/}4+o(1))\left(\genfrac{}{}{0px}{}{n}{3}\right)$ edges. We disprove this conjecture by constructing counterexamples with edge density at least $0.250000356>1\mathrm{/}4$ for all sufficiently large $n$.