Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellens\"atze

TL;DR

This paper explores Ramsey numbers using polynomial ideals and Nullstellensatz, providing algebraic encodings, certificates, and generalizations to other combinatorial numbers, offering new algebraic tools for Ramsey theory.

Contribution

It introduces polynomial encodings for Ramsey graphs, constructs Nullstellensatz certificates linked to online Ramsey numbers, and generalizes these methods to other combinatorial numbers.

Findings

Polynomial encodings for Ramsey graphs

Nullstellensatz certificates with degrees related to online Ramsey numbers

Generalization to Rado, van der Waerden, and Hales-Jewett numbers

Abstract

In this article we study the Ramsey numbers $R(r,s)$ through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions correspond to Ramsey graphs of order $n$, those that do not contain a copy of $K_r$ or $\bar{K}_s$. When these systems have no solution and $n \ge R(r,s)$, we construct Nullstellensatz certificates whose degrees are equal to the restricted online Ramsey numbers introduced by Conlon, Fox, Grinshpun and He. Moreover, we show that these results generalize to other numbers in Ramsey theory, including Rado, van der Waerden, and Hales-Jewett numbers. Finally, we introduce a family of numbers that relate to the coefficients of a certain "Ramsey polynomial" that gives lower bounds for Ramsey numbers.