On efficient constructions of short lists containing mostly Ramsey graphs

TL;DR

This paper explores the construction of short lists of graphs where most are Ramsey graphs, using hardness assumptions to enable polynomial-time algorithms for generating such lists.

Contribution

It introduces a method to efficiently construct short lists of graphs with most being Ramsey graphs, under certain hardness assumptions.

Findings

Polynomial-time construction of lists with mostly Ramsey graphs

Most graphs in the list are 2 log n-Ramsey

Relies on a reasonable hardness assumption

Abstract

One of the earliest and best-known application of the probabilistic method is the proof of existence of a 2 log n$-Ramsey graph, i.e., a graph with n nodes that contains no clique or independent set of size 2 log n. The explicit construction of such a graph is a major open problem. We show that a reasonable hardness assumption implies that in polynomial time one can construct a list containing polylog(n) graphs such that most of them are 2 log n-Ramsey.