How connectivity affects the extremal number of trees

Abstract

The Erdős-Sós conjecture states that the maximum number of edges in an n-vertex graph without a given k-vertex tree is at most $\frac{n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a k-vertex tree T, we construct n-vertex connected graphs that are T-free with at least $(1/4-o_k(1\left)\right)nk$ edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of k-vertex brooms T such that the maximum size of an n-vertex connected T-free graph is at most $(1/4+o_k(1\left)\right)nk$.