Combinatorica最新文献

{"title":"Turán Problems for Expanded Hypergraphs","authors":"Peter Keevash, Noam Lifshitz, Eoin Long, Dor Minzer","doi":"10.1007/s00493-025-00152-4","DOIUrl":"https://doi.org/10.1007/s00493-025-00152-4","url":null,"abstract":"<p>We obtain new results on the Turán number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in Extremal Combinatorics. Firstly, we give general conditions under which the crosscut parameter asymptotically determines the Turán number, thus answering a question of Mubayi and Verstraëte. Secondly, we refine our asymptotic results to obtain several exact results, including proofs of the Huang–Loh–Sudakov conjecture on cross matchings and the Füredi–Jiang–Seiver conjecture on path expansions. We have introduced two major new tools for the proofs of these results. The first of these, Global Hypercontractivity, is used as a ‘black box’ (we present it in a separate paper with several other applications). The second tool, presented in this paper, is a far-reaching extension of the Junta Method, which we develop from a powerful and general technique for finding matchings in hypergraphs under certain pseudorandomness conditions.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"65 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143862129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ordering Candidates via Vantage Points","authors":"Noga Alon, Colin Defant, Noah Kravitz, Daniel G. Zhu","doi":"10.1007/s00493-025-00148-0","DOIUrl":"https://doi.org/10.1007/s00493-025-00148-0","url":null,"abstract":"<p>Given an <i>n</i>-element set <span>(Csubseteq mathbb {R}^d)</span> and a (sufficiently generic) <i>k</i>-element multiset <span>(Vsubseteq mathbb {R}^d)</span>, we can order the points in <i>C</i> by ranking each point <span>(cin C)</span> according to the sum of the distances from <i>c</i> to the points of <i>V</i>. Let <span>(Psi _k(C))</span> denote the set of orderings of <i>C</i> that can be obtained in this manner as <i>V</i> varies, and let <span>(psi ^{textrm{max}}_{d,k}(n))</span> be the maximum of <span>(|Psi _k(C)|)</span> as <i>C</i> ranges over all <i>n</i>-element subsets of <span>(mathbb {R}^d)</span>. We prove that <span>(psi ^{textrm{max}}_{d,k}(n)=Theta _{d,k}(n^{2dk}))</span> when <span>(d ge 2)</span> and that <span>(psi ^{textrm{max}}_{1,k}(n)=Theta _k(n^{4lceil k/2rceil -2}))</span>. As a step toward proving this result, we establish a bound on the number of sign patterns determined by a collection of functions that are sums of radicals of nonnegative polynomials; this can be understood as an analogue of a classical theorem of Warren. We also prove several results about the set <span>(Psi (C)=bigcup _{kge 1}Psi _k(C))</span>; this includes an exact description of <span>(Psi (C))</span> when <span>(d=1)</span> and when <i>C</i> is the set of vertices of a vertex-transitive polytope.\u0000</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"72 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ruzsa’s Problem on Bi-Sidon Sets","authors":"János Pach, Dmitrii Zakharov","doi":"10.1007/s00493-025-00151-5","DOIUrl":"https://doi.org/10.1007/s00493-025-00151-5","url":null,"abstract":"<p>A subset <i>S</i> of real numbers is called <i>bi-Sidon</i> if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of <i>S</i> are distinct. Imre Ruzsa asked the following question: What is the maximum number <i>f</i>(<i>N</i>) such that every set <i>S</i> of <i>N</i> real numbers contains a bi-Sidon subset of size at least <i>f</i>(<i>N</i>)? He proved that <span>(f(N)geqslant cN^{frac{1}{3}})</span>, for a constant <span>(c&gt;0)</span>. In this note, we improve this bound to <span>(N^{frac{1}{3}+frac{7}{78}+o(1)})</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"60 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rigidity Expander Graphs","authors":"Alan Lew, Eran Nevo, Yuval Peled, Orit E. Raz","doi":"10.1007/s00493-025-00149-z","DOIUrl":"https://doi.org/10.1007/s00493-025-00149-z","url":null,"abstract":"<p>Jordán and Tanigawa recently introduced the <i>d</i>-dimensional algebraic connectivity <span>(a_d(G))</span> of a graph <i>G</i>. This is a quantitative measure of the <i>d</i>-dimensional rigidity of <i>G</i> which generalizes the well-studied notion of spectral expansion of graphs. We present a new lower bound for <span>(a_d(G))</span> defined in terms of the spectral expansion of certain subgraphs of <i>G</i> associated with a partition of its vertices into <i>d</i> parts. In particular, we obtain a new sufficient condition for the rigidity of a graph <i>G</i>. As a first application, we prove the existence of an infinite family of <i>k</i>-regular <i>d</i>-rigidity-expander graphs for every <span>(dge 2)</span> and <span>(kge 2d+1)</span>. Conjecturally, no such family of 2<i>d</i>-regular graphs exists. Second, we show that <span>(a_d(K_n)ge frac{1}{2}leftlfloor frac{n}{d}rightrfloor )</span>, which we conjecture to be essentially tight. In addition, we study the extremal values <span>(a_d(G))</span> attains if <i>G</i> is a minimally <i>d</i>-rigid graph.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"37 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143766797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Hypergraph Bipartite Turán Problem with Odd Uniformity","authors":"Jie Ma, Tianchi Yang","doi":"10.1007/s00493-025-00146-2","DOIUrl":"https://doi.org/10.1007/s00493-025-00146-2","url":null,"abstract":"<p>In this paper, we investigate the hypergraph Turán number <span>(textrm{ex}(n,K^{(r)}_{s,t}))</span>. Here, <span>(K^{(r)}_{s,t})</span> denotes the <i>r</i>-uniform hypergraph with vertex set <span>(left( cup _{iin [t]}X_iright) cup Y)</span> and edge set <span>({X_icup {y}: iin [t], yin Y})</span>, where <span>(X_1,X_2,cdots ,X_t)</span> are <i>t</i> pairwise disjoint sets of size <span>(r-1)</span> and <i>Y</i> is a set of size <i>s</i> disjoint from each <span>(X_i)</span>. This study was initially explored by Erdős and has since received substantial attention in research. Recent advancements by Bradač, Gishboliner, Janzer and Sudakov have greatly contributed to a better understanding of this problem. They proved that <span>(textrm{ex}(n,K_{s,t}^{(r)})=O_{s,t}(n^{r-frac{1}{s-1}}))</span> holds for any <span>(rge 3)</span> and <span>(s,tge 2)</span>. They also provided constructions illustrating the tightness of this bound if <span>(rge 4)</span> is <i>even</i> and <span>(tgg sge 2)</span>. Furthermore, they proved that <span>(textrm{ex}(n,K_{s,t}^{(3)})=O_{s,t}(n^{3-frac{1}{s-1}-varepsilon _s}))</span> holds for <span>(sge 3)</span> and some <span>(epsilon _s&gt;0)</span>. Addressing this intriguing discrepancy between the behavior of this number for <span>(r=3)</span> and the even cases, Bradač et al. post a question of whether </p><span>$$begin{aligned} textrm{ex}(n,K_{s,t}^{(r)})= O_{r,s,t}(n^{r-frac{1}{s-1}- varepsilon }) text{ holds } text{ for } text{ odd } rge 5 text{ and } text{ any } sge 3text{. } end{aligned}$$</span><p>In this paper, we provide an affirmative answer to this question, utilizing novel techniques to identify regular and dense substructures. This result highlights a rare instance in hypergraph Turán problems where the solution depends on the parity of the uniformity.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"29 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143713066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved Integrality Gap in Max–Min Allocation, or, Topology at the North Pole","authors":"Penny Haxell, Tibor Szabó","doi":"10.1007/s00493-025-00141-7","DOIUrl":"https://doi.org/10.1007/s00493-025-00141-7","url":null,"abstract":"<p>In the max–min allocation problem a set <i>P</i> of players are to be allocated disjoint subsets of a set <i>R</i> of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezáková and Dani (SIGecom Exch 5(3):11–18, 2005) showed that this problem is NP-hard to approximate within a factor less than 2, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko (Proceedings of the 38th ACM Symposium on Theory of Computing, 2006). Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell (Graphs Comb 11(3):245–248, 1995) for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods, to replace the combinatorial argument of Haxell (Graphs Comb 11(3):245–248, 1995) for the restricted max–min allocation problem. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of 3.808 to 3.534. We also study the <span>((1,varepsilon ))</span>-restricted version, in which resources can take only two values, and improve the integrality gap in most cases. Our approach applies a criterion of Aharoni and Haxell, and Meshulam, for the existence of independent transversals in graphs, which involves the connectedness of the independence complex. This is complemented by a graph process of Meshulam that decreases the connectedness of the independence complex in a controlled fashion and hence, tailored appropriately to the problem, can verify the criterion. In our applications we aim to establish the flexibility of the approach and hence argue for it to be a potential asset in other optimization problems involving hypergraph matchings.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"61 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143713067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Induced Subgraphs of $$K_r$$ -Free Graphs and the Erdős–Rogers Problem","authors":"Lior Gishboliner, Oliver Janzer, Benny Sudakov","doi":"10.1007/s00493-025-00147-1","DOIUrl":"https://doi.org/10.1007/s00493-025-00147-1","url":null,"abstract":"<p>For two graphs <i>F</i>, <i>H</i> and a positive integer <i>n</i>, the function <span>(f_{F,H}(n))</span> denotes the largest <i>m</i> such that every <i>H</i>-free graph on <i>n</i> vertices contains an <i>F</i>-free induced subgraph on <i>m</i> vertices. This function has been extensively studied in the last 60 years when <i>F</i> and <i>H</i> are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic study of this function in the case where <i>F</i> is a general graph. Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer <i>r</i> and every <span>(K_{r-1})</span>-free graph <i>F</i>, there exists some <span>(varepsilon _F&gt;0)</span> such that <span>(f_{F,K_r}(n)=O(n^{1/2-varepsilon _F}))</span>. This result is tight in two ways. Firstly, it is no longer true if <i>F</i> contains <span>(K_{r-1})</span> as a subgraph. Secondly, we show that for all <span>(rge 4)</span> and <span>(varepsilon &gt;0)</span>, there exists a <span>(K_{r-1})</span>-free graph <i>F</i> for which <span>(f_{F,K_r}(n)=Omega (n^{1/2-varepsilon }))</span>. Along the way of proving this, we show in particular that for every graph <i>F</i> with minimum degree <i>t</i>, we have <span>(f_{F,K_4}(n)=Omega (n^{1/2-6/sqrt{t}}))</span>. This answers (in a strong form) another question of Mubayi and Verstraëte. Finally, we prove that there exist absolute constants <span>(0&lt;c&lt;C)</span> such that for each <span>(rge 4)</span>, if <i>F</i> is a bipartite graph with sufficiently large minimum degree, then <span>(Omega (n^{frac{c}{log r}})le f_{F,K_r}(n)le O(n^{frac{C}{log r}}))</span>. This shows that for graphs <i>F</i> with large minimum degree, the behaviour of <span>(f_{F,K_r}(n))</span> is drastically different from that of the corresponding off-diagonal Ramsey number <span>(f_{K_2,K_r}(n))</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"57 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143713068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Large Family of Strongly Regular Graphs with Small Weisfeiler-Leman Dimension","authors":"Jinzhuan Cai, Jin Guo, Alexander L. Gavrilyuk, Ilia Ponomarenko","doi":"10.1007/s00493-025-00145-3","DOIUrl":"https://doi.org/10.1007/s00493-025-00145-3","url":null,"abstract":"<p>In 2002, D. Fon-Der-Flaass constructed a prolific family of strongly regular graphs. In this paper, we prove that for infinitely many natural numbers <i>n</i> and a positive constant <i>c</i>, this family contains at least <span>(n^{ccdot n^{2/3}})</span> strongly regular <i>n</i>-vertex graphs <i>X</i> with the same parameters, which satisfy the following condition: an isomorphism between <i>X</i> and any other graph can be verified by the 4-dimensional Weisfeiler-Leman algorithm.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"28 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143678023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Signed Varchenko Determinant for Complexes of Oriented Matroids","authors":"Winfried Hochstättler, Sophia Keip, Kolja Knauer","doi":"10.1007/s00493-025-00138-2","DOIUrl":"https://doi.org/10.1007/s00493-025-00138-2","url":null,"abstract":"<p>We generalize the (signed) Varchenko matrix of a hyperplane arrangement to complexes of oriented matroids and show that its determinant has a nice factorization. This extends previous results on hyperplane arrangements and oriented matroids.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"33 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143640816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved Lower Bound Towards Chen–Chvátal Conjecture","authors":"Congkai Huang","doi":"10.1007/s00493-025-00137-3","DOIUrl":"https://doi.org/10.1007/s00493-025-00137-3","url":null,"abstract":"<p>We prove that in every metric space where no line contains all the points, there are at least <span>(Omega (n^{2/3}))</span> lines. This improves the previous <span>(Omega (sqrt{n}))</span> lower bound on the number of lines in general metric space, and also improves the previous <span>(Omega (n^{4/7}))</span> lower bound on the number of lines in metric spaces generated by connected graphs.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"86 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}