论文标题
可靠阈值的工具包
A Toolkit for Robust Thresholds
论文作者
论文摘要
考虑一个由于最低度的考虑，它包含一个跨度结构的主机超graph $ g $。我们收集三个结果，证明，如果以适当的速度对$ g $采样的边缘，那么跨度结构仍然以很高的概率出现在采样的超图中。我们证明了在狄拉克阈值上方的超图中的完美匹配，以$ k_r $的图形匹配，以满足hajnal-szemerédi最低度条件的图形和有限度跨度的树木。在每种情况下，我们的证明都是基于构建传播度量，然后在（分数）Kahn-KAlai猜想中应用最新结果，该猜想将这种测量的存在与概率阈值联系起来。 对于我们的第二个结果，我们给出了艾伦（Allen），伯特（Böttcher），科斯滕（Corsten），戴维斯（Corsten），戴维斯（Davies），詹森（Jenssen），莫里斯（Morris），罗伯茨（Roberts）和斯科坎（Skokan）的最新定理的简短和更一般的证明，该定理用不同的技术处理$ r = 3 $ case。特别是，我们回答了他们的问题，即满足hajnal-szemerédi最低度条件的$ k_r $ fextor的数量。
Consider a host hypergraph $G$ which contains a spanning structure due to minimum degree considerations. We collect three results proving that if the edges of $G$ are sampled at the appropriate rate then the spanning structure still appears with high probability in the sampled hypergraph. We prove such results for perfect matchings in hypergraphs above Dirac thresholds, for $K_r$-factors in graphs satisfying the Hajnal--Szemerédi minimum degree condition, and for bounded-degree spanning trees. In each case our proof is based on constructing a spread measure and then applying recent results on the (fractional) Kahn--Kalai conjecture connecting the existence of such measures with probabilistic thresholds. For our second result we give a shorter and more general proof of a recent theorem of Allen, Böttcher, Corsten, Davies, Jenssen, Morris, Roberts, and Skokan which handles the $r=3$ case with different techniques. In particular, we answer a question of theirs with regards to the number of $K_r$-factors in graphs satisfying the Hajnal--Szemerédi minimum degree condition.
