Down‐set thresholds

IF 0.9 3区数学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Random Structures & Algorithms Pub Date : 2021-12-15 DOI:10.1002/rsa.21148

Benjamin Gunby, Xiaoyu He, Bhargav P. Narayanan

引用次数: 1

Abstract

We elucidate the relationship between the threshold and the expectation‐threshold of a down‐set. Qualitatively, our main result demonstrates that there exist down‐sets with polynomial gaps between their thresholds and expectation‐thresholds; in particular, the logarithmic gap predictions of Kahn–Kalai and Talagrand (recently proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about up‐sets do not apply to down‐sets. Quantitatively, we show that any collection 𝒢 of graphs on [n] that covers the family of all triangle‐free graphs on [n] satisfies the inequality ∑G∈𝒢exp(−δe(Gc)/n)<1/2 for some universal δ>0 , and this is essentially best‐possible.

查看原文本刊更多论文

下来量设定阈值

我们阐明了阈值与下集的期望阈值之间的关系。定性地说，我们的主要结果表明，在阈值和期望阈值之间存在多项式差距的下集;特别是，Kahn-Kalai和Talagrand关于上集的对数间隙预测(最近由Park-Pham和frankton - kahn - narayanan - park证明)不适用于下集。定量地，我们证明了[n]上的图的任何集合𝒢覆盖了[n]上的所有无三角形图族，满足不等式∑G∈𝒢exp(−δe(Gc)/n)0，这本质上是最佳可能的。

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

Random Structures & Algorithms 数学-计算机：软件工程

CiteScore

2.50

自引率

10.00%

发文量

审稿时长

>12 weeks

期刊介绍： It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness. Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.