The Tur\'an number of the Cartesian product of trees via star-flip
Motivated by Erd\H{o}s's conjecture on the Tur\'an number of degenerate bipartite graphs, Brada\v{c}, Janzer, Sudakov and Tomon proved that $ \ex(n,T \Box P)=\Theta_{T,P}(n^{3/2})$ for every nontrivial tree $T$ and every nontrivial path $P$, and conjectured that the same order of magnitude holds for the Cartesian product of any two nontrivial trees. We prove their conjecture. More generally, for every integer $r\ge2$, we introduce a class of bipartite $r$-degenerate graphs, called $r$-star-flip graphs, that are obtained from a seed tree by a sequence of local vertex-duplication operations. We prove that every fixed $r$-star-flip graph $H$ satisfies $\ex(n,H)=O_H(n^{2-1/r})$. Every Cartesian product of two trees is a $2$-star-flip graph, while the star-flip class also contains graphs that do not arise as such products. As a further application, our framework yields a new proof of F\"uredi's theorem: if $H$ is a fixed bipartite graph in which at most one vertex in one colour class has degree greater than $r$, then $\ex(n,H)=O_H(n^{2-1/r})$. The key ingredient is a conditional-resampling procedure that extends the tree branching random walk on the seed tree to a random homomorphism of the entire star-flip graph, while preserving the branching-random-walk distribution on every live tree.