Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze

TL;DR

Ramsey numbers are notoriously difficult to determine; the paper develops algebraic encodings to study them. The work presents two complementary approaches: Hilbert's Nullstellensatz-based upper bounds via zero-dimensional ideals and a Combinatorial Nullstellensatz-based lower-bound method using a Ramsey polynomial $f_{r,s,n}$. It introduces new constructs such as the Ramsey-type encodings $RI(n,r,s)$, restricted online Ramsey numbers, and ensemble numbers $E_{n,k,r,H}$, and proves general theorems linking certificate degrees to Ramsey bounds (e.g., degree bounds related to restricted online Ramsey numbers). The framework extends to other Ramsey-type numbers (Rado, van der Waerden, Hales-Jewett) and highlights connections between algebraic certificates and combinatorial games, offering a unified viewpoint on Ramsey numbers and their computational aspects.

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.

Figures (2)

• Figure 1: Builder-Painter game for $R(3,3)$. For each turn, the edge selected by Builder is indicated in black, and Painter colors it either red or blue.
• Figure 2: Valid coverings of the graph $H$ with edges $(1,2)$ and $(1,5)$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Definition 3
• Theorem 4
• Theorem 5
• proof : Proof of Theorem \ref{['binEdge2']}
• Example 6
• proof
• proof : Proof of Theorem \ref{['RamseyNullUpperBound']}
• proof : Proof of Theorem \ref{['NullDegree_general_case']}