For t by t split patterns, the new upper bound is n^{2 - 1/t + o(1)}, and for one-sided t-split patterns it is n^{2 - 1/t + 1/(2t^2) + o(1)}.
The Tur\'an number of blow-ups of trees
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abstract
A conjecture of Erd\H{o}s from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this bound holds for a large family of $r$-degenerate bipartite graphs, including all $r$-degenerate blow-ups of trees. Our results generalise many previously proven cases of the Erd\H{o}s conjecture, including the related results of F\"uredi and Alon, Krivelevich and Sudakov. Our proof uses supersaturation and a random walk on an auxiliary graph.
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Bipartite Tur\'an problems for ordered graphs
For t by t split patterns, the new upper bound is n^{2 - 1/t + o(1)}, and for one-sided t-split patterns it is n^{2 - 1/t + 1/(2t^2) + o(1)}.