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)}.
Erd˝ os, On extremal problems of graphs and generalized graphs, Israel J
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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)}.