论文标题
部分可观测时空混沌系统的无模型预测
New Ramsey Multiplicity Bounds and Search Heuristics
论文作者
论文摘要
我们研究了两个相关的问题，这些问题涉及图表中给定大小的均质子集的数量，这些问题可以追溯到ERDS的问题。最值得注意的是，我们改善了Ramsey多重性的上限和$ K_4 $和$ K_5 $，并结算了最低尺寸$ 4 $的独立数量，其中包含$ 4 $。由对称的Ramsey多样性问题的令人震惊的动机，我们还引入了一个偏离的变体，并在对单色$ k_4 $或$ k_5 $计数时仅在另一种颜色和三角形中计数$ k_4 $或$ k_5 $。每个问题的极端构造结果都是恒定尺寸的爆炸，是通过搜索启发式方法找到的。它们与使用FLAG代数建立的下限相辅相成，从而实现了完全计算机辅助的方法。对于我们的某些定理，我们还可以得出极端构造在非常强烈的意义上是稳定的。更广泛地说，这些问题使我们研究了可能成对和独立集的密度，这些区域可以被视为某些图的极限。
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.
