The Turán number of the grid

For a positive integer $t$, let $F_t$ denote the graph of the $t\times t$ grid. Motivated by a 50-year-old conjecture of Erdős about Turán numbers of $r$-degenerate graphs, we prove that there exists a constant $C=C(t)$ such that $\mathrm{ex}(n,F_t)\leq Cn^{3/2}$. This bound is tight up to the value of $C$. One of the interesting ingredients of our proof is a novel way of using the tensor power trick.

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