The Ramsey number is the minimum number of nodes, $n = R(s, t)$, such that all undirected simple graphs of order $n$, contain a clique of order $s$, or an independent set of order $t$. This paper explores the application of a best first search algorithm and reinforcement learning (RL) techniques to find counterexamples to specific Ramsey numbers. We incrementally improve over prior search methods such as random search by introducing a graph vectorization and deep neural network (DNN)-based heuristic, which gauge the likelihood of a graph being a counterexample. The paper also proposes algorithmic optimizations to confine a polynomial search runtime. This paper does not aim to present new counterexamples but rather introduces and evaluates a framework supporting Ramsey counterexample exploration using other heuristics. Code and methods are made available through a PyPI package and GitHub repository.

翻译：拉姆齐数定义为最小节点数$n = R(s, t)$，使得所有$n$阶无向简单图必包含$s$阶团或$t$阶独立集。本文探究最佳优先搜索算法与强化学习(RL)技术在特定拉姆齐数反例发现中的应用。通过引入图向量化及基于深度神经网络(DNN)的启发函数（用于评估图成为反例的可能性），我们逐步改进了随机搜索等现有搜索方法。本文还提出算法优化策略以限定多项式搜索运行时。本研究并非旨在展示新反例，而是介绍并评估一个支持基于其他启发函数探索拉姆齐反例的框架。相关代码与方法已通过PyPI包及GitHub仓库公开提供。