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.

我们研究了两个相关问题，涉及图中给定大小的同质子集的数量，这可以追溯到 Erdős 的问题。最值得注意的是，我们改进了\(K_4\)和\(K_5\)的拉姆齐重数的上限，并确定了团数最多为 4 的图中大小为 4 的独立集的最小数量。为了解决对称 Ramsey 多重性问题，我们还引入了非对角变体，并在仅计算一种颜色中的单色\(K_4\)或\(K_5\)和另一种颜色中的三角形时获得严格的结果。每个问题的极值结构都是恒定大小的图的放大图，并通过搜索启发法找到。它们由使用标志代数建立的下界进行补充，从而形成完全计算机辅助的方法。对于我们的一些定理，我们还可以得出极值结构在很强的意义上是稳定的。更广泛地说，这些问题引导我们研究可能的派系和独立集密度对的区域，这些密度可以作为某些图序列的极限来实现。