新的拉姆齐多重性边界和搜索启发法

IF 2.5 1区 数学 Q2 COMPUTER SCIENCE, THEORY & METHODS
Olaf Parczyk, Sebastian Pokutta, Christoph Spiegel, Tibor Szabó
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引用次数: 0

摘要

我们研究了关于图中给定大小的同质子集数的两个相关问题,这些问题可以追溯到厄尔多斯(Erdős)的问题。最值得注意的是,我们改进了 \(K_4\) 和 \(K_5\) 的拉姆齐多重性的上界,并解决了具有最多 4 个簇数的图中大小为 4 的独立集的最小数量问题。受对称拉姆齐多重性问题难以捉摸的启发,我们还引入了非对角线变体,并在只计算一种颜色的单色(K_4)或(K_5)和另一种颜色的三角形时得到了严密的结果。每个问题的极值构造都是通过搜索启发式发现的恒定大小图的吹胀。此外,我们还利用旗标代数建立了下界,从而形成了一种完全由计算机辅助的方法。对于我们的某些定理,我们还可以推导出极值构造在非常强的意义上是稳定的。从更广泛的意义上讲,这些问题引导我们研究可以作为某些图序列的极限来实现的可能的簇和独立集密度对的区域。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

New Ramsey Multiplicity Bounds and Search Heuristics

New Ramsey Multiplicity Bounds and Search Heuristics

We study two related problems concerning the number of homogeneous subsets of given size in graphs that go back to questions of Erdős. Most notably, we improve the upper bounds on the Ramsey multiplicity of \(K_4\) and \(K_5\) and settle the minimum number of independent sets of size 4 in graphs with clique number at most 4. Motivated by the elusiveness of the symmetric Ramsey multiplicity problem, we also introduce an off-diagonal variant and obtain tight results when counting monochromatic \(K_4\) or \(K_5\) in only one of the colors and triangles in the other. The extremal constructions for each problem turn out to be blow-ups of a graph of constant size and were found through search heuristics. They are complemented by lower bounds established using flag algebras, resulting in a fully computer-assisted approach. For some of our theorems we can also derive that the extremal construction is stable in a very strong sense. More broadly, these problems lead us to the study of the region of possible pairs of clique and independent set densities that can be realized as the limit of some sequence of graphs.

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来源期刊
Foundations of Computational Mathematics
Foundations of Computational Mathematics 数学-计算机:理论方法
CiteScore
6.90
自引率
3.30%
发文量
46
审稿时长
>12 weeks
期刊介绍: Foundations of Computational Mathematics (FoCM) will publish research and survey papers of the highest quality which further the understanding of the connections between mathematics and computation. The journal aims to promote the exploration of all fundamental issues underlying the creative tension among mathematics, computer science and application areas unencumbered by any external criteria such as the pressure for applications. The journal will thus serve an increasingly important and applicable area of mathematics. The journal hopes to further the understanding of the deep relationships between mathematical theory: analysis, topology, geometry and algebra, and the computational processes as they are evolving in tandem with the modern computer. With its distinguished editorial board selecting papers of the highest quality and interest from the international community, FoCM hopes to influence both mathematics and computation. Relevance to applications will not constitute a requirement for the publication of articles. The journal does not accept code for review however authors who have code/data related to the submission should include a weblink to the repository where the data/code is stored.
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