Small dense subgraphs of polarity graphs and the extremal number for the 4-cycle

TL;DR

This paper investigates small dense subgraphs within polarity graphs derived from projective planes, providing new lower bounds for the extremal number of 4-cycles and disproving a related conjecture.

Contribution

It introduces a construction of small dense subgraphs in polarity graphs and establishes improved bounds on the Turán number for 4-cycles, challenging previous conjectures.

Findings

Identifies dense subgraphs in polarity graphs with specific vertex and edge counts.

Provides improved lower bounds for the extremal number ex(n, C_4).

Disproves a conjecture regarding ex(q^2 - q - 2, C_4) for q a power of 2.

Abstract

In this note, we show that for any $m \in \{1,2, \dots , q +1 \}$, if $G$ is a polarity graph of a projective plane of order $q$ that has an oval, then $G$ contains a subgraph on $m + \binom{m}{2}$ vertices with $m^2+\frac{m^4}{8q} - O ( \frac{m^4}{q^{3/2} } +m )$ edges. As an application, we give the best known lower bounds on the Tur\'{a}n number $\mathrm{ex}(n, C_4)$ for certain values of $n$. In particular, we disprove a conjecture of Abreu, Balbuena, and Labbate concerning $\mathrm{ex}(q^2-q-2, C_4)$ where $q$ is a power of $2$.