Paley-like graphs for the Ramsey number $r(C_4,K_t)$

TL;DR

This paper discusses the construction of specific $C_4$-free graphs related to the Ramsey number $r(C_4,K_t)$, addressing a conjecture about their independence number and clarifying its falsehood.

Contribution

The paper presents a construction of $C_4$-free graphs and refutes a previous conjecture regarding their independence number.

Findings

The conjecture about the independence number being $n^{1/2+o(1)}$ is false.

Constructed graphs are $C_4$-free and related to the Ramsey number $r(C_4,K_t)$.

The paper clarifies the limitations of the earlier conjecture.

Abstract

An earlier version of this paper constructed a family of $n$-vertex $C_4$-free graphs which we conjectured to have independence number $n^{\frac 12+o(1)}$. This conjecture is false, as pointed out by Michael Tait.