Bounds on Small Ramsey Numbers by Semidefinite Programming

IF 8.2 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Review Pub Date : 2026-05-13 DOI:10.1137/25m1827554

Bernard Lidický, Florian Pfender

引用次数: 0

Abstract

SIAM Review, Volume 68, Issue 2, Page 385-403, June 2026.
Abstract.Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method introduced by Razborov in 2007 was developed to find asymptotic results for very large graphs, so it seems that the method should not be suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove several exact values. The main power of the method relies on utilizing semidefinite programming to find certificates that are sums of squares. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.

查看原文本刊更多论文

半定规划下小Ramsey数的界

SIAM评论，68卷，第2期，385-403页，2026年6月。摘要。寻找精确的拉姆齐数是一个通常局限于相对较小的图的问题。2007年Razborov引入的标志代数方法是为了寻找非常大的图的渐近结果而发展起来的，因此该方法似乎不适合寻找小的Ramsey数。但这种直觉是错误的，我们将在本文中开发一种技术来做到这一点。我们找到了许多小图和超图拉姆齐数的新的上界。因此，我们证明了几个精确值。该方法的主要功能依赖于利用半定规划来查找平方和的证书。我们希望这一技术将适用于使用标志代数方法解决较小图的其他问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Review 数学-应用数学

CiteScore

16.90

自引率

0.00%

发文量

期刊介绍： Survey and Review feature papers that provide an integrative and current viewpoint on important topics in applied or computational mathematics and scientific computing. These papers aim to offer a comprehensive perspective on the subject matter. Research Spotlights publish concise research papers in applied and computational mathematics that are of interest to a wide range of readers in SIAM Review. The papers in this section present innovative ideas that are clearly explained and motivated. They stand out from regular publications in specific SIAM journals due to their accessibility and potential for widespread and long-lasting influence.