Tur\'an number of bipartite graphs with no $K_{t,t}$

TL;DR

This paper proves that bipartite graphs with bounded degree in one part and no $K_{t,t}$ subgraph have extremal numbers growing slower than the Kővári-Sós-Turán bound, confirming a conjecture by Conlon, Janzer, and Lee.

Contribution

It establishes a new upper bound on extremal numbers for a class of bipartite graphs, verifying a conjecture and extending understanding of extremal graph theory.

Findings

Extremal number for certain bipartite graphs is o(n^{2-1/t})

Confirms a conjecture by Conlon, Janzer, and Lee

Generalizes bounds for bipartite graphs with degree constraints

Abstract

The extremal number of a graph $H$, denoted by $\mbox{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices that does not contain $H$. The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem says that for a complete bipartite graph with parts of size $t\leq s$ the extremal number is $\mbox{ex}(K_{s,t})=O(n^{2-1/t})$. It is also known that this bound is sharp if $s>(t-1)!$. In this paper, we prove that if $H$ is a bipartite graph such that all vertices in one of its parts have degree at most $t$, but $H$ contains no copy of $K_{t,t}$, then $\mbox{ex}(n,H)=o(n^{2-1/t})$. This verifies a conjecture of Conlon, Janzer and Lee.