As the title suggests: how many edges (as a function of $n$) can a bipartite graph with $n$ nodes on each side have if it does not have $K_{4, 4}$ as a subgraph? Is this known?
2026-03-26 14:25:14.1774535114
What is the densest balanced bipartite graph without any $K_{4, 4}$?
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Quite surprisingly (at least to me) the general case seems to be an open problem, the Zarankiewicz problem. However, some estimates are due to Kővári–Sós–Turán, and maybe the specific $K_{4,4}$ case is entirely solved. I am afraid I am not able to be more helpful than this (i.e. very little).