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Coordinates","{\"openalex\":\"W1986296546\",\"doi\":\"10.1007/pl00009809\"}","2026-05-22T11:25:02.245187277","2026-05-22T11:25:02.245188772","doi","10.1007/pl00009809","arxivId","openalexId","W1986296546","abstractText","publishedDate","1998-01-01","publishedYear",1998,"venue","COMBINATORICA",249,"authors","keywords",[189,190,191],"pdfUrl","topVenue","Rough Sets and Fuzzy Logic","Digital Image Processing Techniques","Complexity and Algorithms in Graphs","paper_W1529556207","Every monotone graph property has a sharp threshold","{\"openalex\":\"W1529556207\",\"doi\":\"10.1090/s0002-9939-96-03732-x\"}","2026-05-16T16:27:44.255926051","2026-05-19T08:03:50.181764815","10.1090/s0002-9939-96-03732-x","W1529556207","In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let V n ( p ) = { 0 , 1 } n V_n(p)= \\{0,1\\}^n denote the Hamming space endowed with the probability measure μ p \\mu _p defined by μ p ( ϵ 1 , ϵ 2 , … , ϵ n ) = p k ⋅ ( 1 − p ) n − k \\mu _p (\\epsilon _1, \\epsilon _2, \\dots , \\epsilon _n)= p^k \\cdot (1-p)^{n-k} , where k = ϵ 1 + ϵ 2 + ⋯ + ϵ n k=\\epsilon _1 +\\epsilon _2 +\\cdots +\\epsilon _n . Let A A be a monotone subset of V n V_n . We say that A A is symmetric if there is a transitive permutation group Γ \\Gamma","1996-10-01",1996,"Proceedings of the American Mathematical Society",435,[206,207,208],"https://www.ams.org/proc/1996-124-10/S0002-9939-96-03732-X/S0002-9939-96-03732-X.pdf","Limits and Structures in Graph Theory","Advanced Graph Theory Research","Stochastic processes and statistical mechanics","paper_W1559492868","Sharp thresholds of graph properties, and the $k$-sat problem","{\"openalex\":\"W1559492868\",\"doi\":\"10.1090/s0894-0347-99-00305-7\"}","2026-05-15T05:28:29.21119425","2026-05-21T03:52:32.195525124","10.1090/s0894-0347-99-00305-7","W1559492868","Given a monotone graph property $P$, consider $\\mu _p(P)$, the probability that a random graph with edge probability $p$ will have $P$. The function $d\\mu _p(P)/dp$ is the key to understanding the threshold behavior of the property $P$. We show that if $d\\mu _p(P)/dp$ is small (corresponding to a non-sharp threshold), then there is a list of graphs of bounded size such that $P$ can be approximated by the property of having one of the graphs as a subgraph. One striking consequence of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of $n$. As an application of the main theorem we settle the question of the existence of a sharp threshold for the satisfiability of a random $k$-CNF formula. An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix some of the conjectures raised in this paper are proven, along with more general results.","1999-05-27",1999,"Journal of the American Mathematical Society",691,[206,207,191],"https://www.ams.org/jams/1999-12-04/S0894-0347-99-00305-7/S0894-0347-99-00305-7.pdf",{"_59":224,"_61":225},[],[226],{"_64":227,"_66":228,"_68":229,"_70":230},1784585775786,{"_73":232,"_75":76,"_77":227,"_78":-5,"_79":26,"_80":26,"_81":26,"_82":-5,"_83":-5,"_84":85,"_35":86,"_87":88},[72,231],"[\"portal\",\"industries\"]","industries",[233,234,235,236,237,238,239,240,241,242,243,244],{"_15":306,"_29":307,"_31":-5,"_32":308,"_33":309,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":310,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":307,"_58":309},{"_15":302,"_29":303,"_31":-5,"_32":-5,"_33":304,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":305,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":303,"_58":304},{"_15":296,"_29":297,"_31":-5,"_32":298,"_33":299,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":300,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":297,"_58":299},{"_15":292,"_29":293,"_31":-5,"_32":-5,"_33":294,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":295,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":293,"_58":294},{"_15":288,"_29":289,"_31":-5,"_32":-5,"_33":290,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":291,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":289,"_58":290},{"_15":282,"_29":283,"_31":-5,"_32":284,"_33":285,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":286,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":283,"_58":285},{"_15":277,"_29":278,"_31":-5,"_32":-5,"_33":279,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":280,"_257":26,"_258":281,"_259":26,"_260":-5,"_57":278,"_58":279},{"_15":273,"_29":274,"_31":-5,"_32":-5,"_33":275,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":276,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":274,"_58":275},{"_15":269,"_29":270,"_31":-5,"_32":-5,"_33":271,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":272,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":270,"_58":271},{"_15":265,"_29":266,"_31":-5,"_32":-5,"_33":267,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":268,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":266,"_58":267},{"_15":261,"_29":262,"_31":-5,"_32":-5,"_33":263,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":264,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":262,"_58":263},{"_15":245,"_29":246,"_31":-5,"_32":-5,"_33":247,"_34":-5,"_35":36,"_37":-5,"_39":-5,"_40":248,"_42":-5,"_44":-5,"_249":250,"_251":-5,"_252":76,"_253":254,"_255":256,"_257":26,"_258":85,"_259":26,"_260":-5,"_57":246,"_58":247},"tax_ind_smartphone","智能手机","Smartphone","manual","taxonomyType","INDUSTRY","code","level","parentId","tax_ind_root","path","科技行业/智能手机","sortOrder","isLeaf","entityCount","children","tax_ind_robot","机器人","Robotics","科技行业/机器人","tax_ind_wearable","智能穿戴","Smart Wearables","科技行业/智能穿戴","tax_ind_energy","新能源与智能制造","New Energy & Smart Manufacturing","科技行业/新能源与智能制造","tax_ind_biotech","生物科技","Biotech","科技行业/生物科技","tax_ind_quantum","量子计算","Quantum Computing","科技行业/量子计算",true,"tax_ind_blockchain","区块链",[287],"Blockchain","科技行业/区块链","Web3","tax_ind_enterprise","企业软件","Enterprise Software","科技行业/企业软件","tax_ind_security","网络安全","Cybersecurity","科技行业/网络安全","tax_ind_semi","半导体",[301],"Semiconductor","科技行业/半导体","芯片","tax_ind_cloud","云计算","Cloud Computing","科技行业/云计算","tax_ind_ai","人工智能",[311,312],"Artificial Intelligence","科技行业/人工智能","AI","AI 技术","env",{"_315":24},"SITE_BASE_URL"]

Ehud Friedgut 人物

Ehud Friedgut · 相关文章