The Turán number of the Cartesian product of graphs

Abstract

Recently, Domagoj Bradač, Oliver Janzer, Benny Sudakov and István Tomon have proved that the Turán number of $2$-dimensional grids is $\Theta (n^{3/2})$, or more general, $\mathrm{ex}\left(n,T\square P\right)=\Theta (n^{3/2})$, where $T$ is a non-trivial tree, $P$ is a non-trivial path, and $T\square P$ denotes the Cartesian product. In their proof, they exhibited a novel way of using the tensor power trick, which has lots of potential in Turán type problems. By the end of their proof, they conjectured that $\mathrm{ex}\left(n,T\square R\right)=\Theta (n^{3/2})$ for non-trivial trees $T$ and $R$. This paper is an extension based on their work, we successfully prove the above conjecture by adapting their approach.

There is a major mistake in my argument that can not be fixed soon