The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$

TL;DR

This paper uses computational algorithms to determine specific Ramsey numbers involving triangles and nearly complete graphs, providing exact values, new bounds, and computational methods for these complex combinatorial problems.

Contribution

The paper establishes the exact value of R(3,K_{10}-e), introduces new bounds for R(3,K_k-e), and computes several previously unknown Ramsey numbers using advanced computational techniques.

Findings

R(3,K_{10}-e)=37 was established.

New upper bounds for R(3,K_k-e) for 11 ≤ k ≤ 16.

Determined R(3,K_{10}-K_3-e)=31 and R(3,K_{10}-P_3-e)=31.

Abstract

Using computer algorithms we establish that the Ramsey number $R(3,K_{10}-e)$ is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of $R(3,K_k-e)$ for $11 \le k \le 16$, and show by construction a new lower bound $55 \le R(3,K_{13}-e)$. The new upper bounds on $R(3,K_k-e)$ are obtained by using the values and lower bounds on $e(3,K_l-e,n)$ for $l \le k$, where $e(3,K_k-e,n)$ is the minimum number of edges in any triangle-free graph on $n$ vertices without $K_k-e$ in the complement. We complete the computation of the exact values of $e(3,K_k-e,n)$ for all $n$ with $k \leq 10$ and for $n \leq 34$ with $k = 11$, and establish many new lower bounds on $e(3,K_k-e,n)$ for higher values of $k$. Using the maximum triangle-free graph generation method, we determine two other previously unknown Ramsey numbers, namely $R(3,K_{10}-K_3-e)=31$ and $R(3,K_{10}-P_3-e)=31$. For graphs $G$ on 10 vertices, %besides $G=K_{10}$, this leaves 6 other open besides $G=K_{10}$, this leaves 6 open cases of the form $R(3,G)$. The hardest among them appears to be $G=K_{10}-2K_2$, for which we establish the bounds $31 \le R(3,K_{10}-2K_2) \le 33$.