Explicit geometric construction of Ramsey graphs

TL;DR

This paper introduces an explicit geometric method to construct large families of triangle-free graphs with bounded independence numbers, providing new bounds on Ramsey numbers and an efficient approximation algorithm for maximum independent sets.

Contribution

It presents a novel geometric construction of Ramsey graphs, offers a new combinatorial proof for independence bounds, and develops a linear-time approximation algorithm for independent sets.

Findings

Constructed explicit triangle-free graphs with independence number O(n^{2/3})

Provided a constructive lower bound of Ω(t^{3/2}) for R(3,t)

Developed a linear-time 1/2-approximation algorithm for maximum independent set

Abstract

We present an explicit geometric construction of a large parametrized family of graphs with no $k$-cliques and with bounded independence number, generalizing the triangle-free Ramsey graphs of Codenotti, Pudl\'ak, and Resta and revisiting the previous generalization by Kostochka, Pudl\'ak and R\"odl. Within this framework, we provide a new combinatorial proof of the upper bound on the independence number for their constructions, offering additional insight into the structure of its independent sets. For our generalized family, we establish lower bounds on the independence number and identify necessary constraints on the parameters under which these graphs could yield improved constructive asymptotic lower bounds on $R(s,t)$, with particular emphasis on $R(3,t)$. As a byproduct, we provide a linear-time approximation algorithm for finding the largest independent set within this parametrized family. For a substantial subfamily, this algorithm achieves a $\frac{1}{2}$-approximation ratio. Special attention is given to the case of triangle-free graphs: we describe explicit constructions on $n$ vertices with independence number $O(n^{\frac{2}{3}})$, giving a constructive asymptotic lower bound of $\Omega(t^{\frac{3}{2}})$ for the Ramsey numbers $R(3,t)$, matching the best known constructive bound.