Spanning trees in graphs without large bipartite holes

Abstract We show that for any $\varepsilon \gt 0$ and $\Delta \in \mathbb{N}$ , there exists $\alpha \gt 0$ such that for sufficiently large $n$ , every $n$ -vertex graph $G$ satisfying that $\delta (G)\geq \varepsilon n$ and $e(X, Y)\gt 0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all spanning trees with maximum degree at most $\Delta$ . This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.