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In recent years, an important trend in the area has been to extend such classical results to the so-called ‘sparse random setting’. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. Although these two articles solved very similar sets of longstanding open problems, the methods used are very different from one another and have different strengths and weaknesses. In this article, we provide a third, completely different, approach to proving extremal and structural results in sparse random sets that also yields their natural ‘counting’ counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the well-known conjecture of Kohayakawa, Łuczak, and Rödl, a probabilistic embedding lemma for sparse graphs. We also give alternative proofs of many of the results of Conlon and Gowers and of Schacht, such as sparse random versions of Szemerédi’s theorem, the Erdős–Stone Theorem, and the Erdős–Simonovits Stability Theorem, and obtain their natural ‘counting’ versions, which in some cases are considerably stronger. For example, we show that for each positive β \\beta and integer k k , there are at most ( β n m ) \\binom {\\beta n}{m} sets of size m m that contain no k k -term arithmetic progression, provided that m ⩾ C n 1 − 1 / ( k − 1 ) m \\geqslant Cn^{1-1/(k-1)} , where C C is a constant depending only on β \\beta and k k . We also obtain new results, such as a sparse version of the Erdős–Frankl–Rödl Theorem on the number of H H -free graphs and, as a consequence of the KŁR conjecture, we extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, non-symmetric setting.","publishedDate","2014-08-07","publishedYear",2014,"venue","Journal of the American Mathematical Society",340,"authors","keywords",[152,153,154],"pdfUrl","https://www.ams.org/jams/2015-28-03/S0894-0347-2014-00816-X/S0894-0347-2014-00816-X.pdf","topVenue","Limits and Structures in Graph Theory","Advanced Graph Theory Research","Graph theory and applications",{"_57":156,"_59":157},[],[158],{"_62":159,"_64":160,"_66":161,"_68":162},1780518497445,{"_71":164,"_73":74,"_75":159,"_76":-5,"_77":78,"_79":78,"_80":78,"_81":-5,"_82":-5,"_83":84,"_33":85,"_86":87},[70,163],"[\"portal\",\"industries\"]","industries",[165,166,167,168,169,170,171,172,173,174,175,176],{"_15":238,"_27":239,"_29":-5,"_30":240,"_31":241,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":242,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":239,"_56":241},{"_15":234,"_27":235,"_29":-5,"_30":-5,"_31":236,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":237,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":235,"_56":236},{"_15":228,"_27":229,"_29":-5,"_30":230,"_31":231,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":232,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":229,"_56":231},{"_15":224,"_27":225,"_29":-5,"_30":-5,"_31":226,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":227,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":225,"_56":226},{"_15":220,"_27":221,"_29":-5,"_30":-5,"_31":222,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":223,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":221,"_56":222},{"_15":214,"_27":215,"_29":-5,"_30":216,"_31":217,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":218,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":215,"_56":217},{"_15":209,"_27":210,"_29":-5,"_30":-5,"_31":211,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":212,"_189":78,"_190":213,"_191":78,"_192":-5,"_55":210,"_56":211},{"_15":205,"_27":206,"_29":-5,"_30":-5,"_31":207,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":208,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":206,"_56":207},{"_15":201,"_27":202,"_29":-5,"_30":-5,"_31":203,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":204,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":202,"_56":203},{"_15":197,"_27":198,"_29":-5,"_30":-5,"_31":199,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":200,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":198,"_56":199},{"_15":193,"_27":194,"_29":-5,"_30":-5,"_31":195,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":196,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":194,"_56":195},{"_15":177,"_27":178,"_29":-5,"_30":-5,"_31":179,"_32":-5,"_33":34,"_35":-5,"_37":-5,"_38":180,"_40":-5,"_42":-5,"_181":182,"_183":-5,"_184":74,"_185":186,"_187":188,"_189":78,"_190":84,"_191":78,"_192":-5,"_55":178,"_56":179},"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",[219],"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",[233],"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",[243,244],"Artificial Intelligence","科技行业/人工智能","AI","AI 技术","env",{"_247":23},"SITE_BASE_URL"]

József Balogh 人物

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