The Tur\'an number of the grid

TL;DR

This paper establishes an upper bound on the Turán number of grid graphs, showing it grows at most on the order of n^{3/2}, and introduces a novel tensor power technique in the proof.

Contribution

It proves a tight upper bound on the Turán number for grid graphs, advancing understanding related to Erdős's conjecture on r-degenerate graphs.

Findings

Bound on ex(n, F_t) is at most C n^{3/2} for some constant C

The bound is tight up to the constant factor

Introduces a new application of the tensor power trick

Abstract

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 Erd\H{o}s about Tur\'{a}n 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.