A construction for bipatite Tur\'an numbers

TL;DR

This paper explores the bipartite Turán numbers, demonstrating that certain graphs $G(q,t)$ provide asymptotic bounds for various complete bipartite graphs, including $K_{3,3}$, matching known bounds from Norm-graphs.

Contribution

It shows that specific constructions $G(q,t)$ can be used to establish asymptotic bounds for multiple bipartite Turán problems, extending their applicability beyond $K_{2,t+1}$.

Findings

$G(q,t)$ graphs give asymptotic bounds for $K_{3,3}$.

$G(q,t)$ graphs match bounds from Norm-graphs.

Extension of bipartite Turán bounds to higher complete bipartite graphs.

Abstract

We consider in detail the well-known family of graphs $G(q,t)$ that establish an asymptotic lower bound for Tur\'an numbers $\mathrm{ex}(n,K_{2,t+1})$. We prove that $G(q,t)$ for some specific $q$ and $t$ also gives an asymptotic bound for $K_{3,3}$ and for some higher complete bipartite graphs as well. The asymptotic bounds we prove are the same as provided by the well-known Norm-graphs.