CombinatoricaPub Date : 2025-03-27DOI: 10.1007/s00493-025-00147-1
Lior Gishboliner, Oliver Janzer, Benny Sudakov
{"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>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 >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<c<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}
CombinatoricaPub Date : 2025-03-24DOI: 10.1007/s00493-025-00145-3
Jinzhuan Cai, Jin Guo, Alexander L. Gavrilyuk, Ilia Ponomarenko
{"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}
CombinatoricaPub Date : 2025-03-17DOI: 10.1007/s00493-025-00138-2
Winfried Hochstättler, Sophia Keip, Kolja Knauer
{"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}
CombinatoricaPub Date : 2025-03-14DOI: 10.1007/s00493-025-00137-3
Congkai Huang
{"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}
CombinatoricaPub Date : 2025-03-14DOI: 10.1007/s00493-025-00142-6
András Mészáros
{"title":"Bounds on the Mod 2 Homology of Random 2-Dimensional Determinantal Hypertrees","authors":"András Mészáros","doi":"10.1007/s00493-025-00142-6","DOIUrl":"https://doi.org/10.1007/s00493-025-00142-6","url":null,"abstract":"<p>As a first step towards a conjecture of Kahle and Newman, we prove that if <span>(T_n)</span> is a random 2-dimensional determinantal hypertree on <i>n</i> vertices, then </p><span>$$begin{aligned} frac{dim H_1(T_n,mathbb {F}_2)}{n^2} end{aligned}$$</span><p>converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the 1-out 2-complex. Our proof relies on the large deviation principle for the Erdős–Rényi random graph by Chatterjee and Varadhan.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"88 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618493","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}
CombinatoricaPub Date : 2025-03-14DOI: 10.1007/s00493-025-00143-5
Jie Ma, Long-Tu Yuan
{"title":"Supersaturation Beyond Color-Critical Graphs","authors":"Jie Ma, Long-Tu Yuan","doi":"10.1007/s00493-025-00143-5","DOIUrl":"https://doi.org/10.1007/s00493-025-00143-5","url":null,"abstract":"<p>The supersaturation problem for a given graph <i>F</i> asks for the minimum number <span>(h_F(n,q))</span> of copies of <i>F</i> in an <i>n</i>-vertex graph with <span>(textrm{ex}(n,F)+q)</span> edges. Subsequent works by Rademacher, Erdős, and Lovász and Simonovits determine the optimal range of <i>q</i> (which is linear in <i>n</i>) for cliques <i>F</i> such that <span>(h_F(n,q))</span> equals the minimum number <span>(t_F(n,q))</span> of copies of <i>F</i> obtained from a maximum <i>F</i>-free <i>n</i>-vertex graph by adding <i>q</i> new edges. A breakthrough result of Mubayi extends this line of research from cliques to color-critical graphs <i>F</i>, and this was further strengthened by Pikhurko and Yilma who established the equality <span>(h_F(n,q)=t_F(n,q))</span> for <span>(1le qle epsilon _F n)</span> and sufficiently large <i>n</i>. In this paper, we present several results on the supersaturation problem that extend beyond the existing framework. Firstly, we explicitly construct infinitely many graphs <i>F</i> with restricted properties for which <span>(h_F(n,q)<qcdot t_F(n,1))</span> holds when <span>(ngg qge 4)</span>, thus refuting a conjecture of Mubayi. Secondly, we extend the result of Pikhurko–Yilma by showing the equality <span>(h_F(n,q)=t_F(n,q))</span> in the range <span>(1le qle epsilon _F n)</span> for any member <i>F</i> in a diverse and abundant graph family (which includes color-critical graphs, disjoint unions of cliques <span>(K_r)</span>, and the Petersen graph). Lastly, we prove the existence of a graph <i>F</i> for any positive integer <i>s</i> such that <span>(h_F(n,q)=t_F(n,q))</span> holds when <span>(1le qle epsilon _F n^{1-1/s})</span>, and <span>(h_F(n,q)<t_F(n,q))</span> when <span>(n^{1-1/s}/epsilon _Fle qle epsilon _F n)</span>, indicating that <span>(q=Theta (n^{1-1/s}))</span> serves as the threshold for the equality <span>(h_F(n,q)=t_F(n,q))</span>. We also discuss some additional remarks and related open problems.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"183 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618494","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}
CombinatoricaPub Date : 2025-03-14DOI: 10.1007/s00493-025-00140-8
David Hartman, Tomáš Hons, Jaroslav Nešetřil
{"title":"Gadget Construction and Structural Convergence","authors":"David Hartman, Tomáš Hons, Jaroslav Nešetřil","doi":"10.1007/s00493-025-00140-8","DOIUrl":"https://doi.org/10.1007/s00493-025-00140-8","url":null,"abstract":"<p>Nešetřil and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field is relatively young, the range of examples of convergent sequences is limited and only a few methods of construction are known. Our aim is to extend the variety of constructions by considering the gadget construction. We show that, when restricting to the set of sentences, the application of gadget construction on elementarily convergent sequences yields an elementarily convergent sequence. On the other hand, we show counterexamples witnessing that a generalization to the full first-order convergence is not possible without additional assumptions. We give several different sufficient conditions to ensure the full convergence. One of them states that the resulting sequence is first-order convergent if the replaced edges are dense in the original sequence of structures.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"56 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618492","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}
CombinatoricaPub Date : 2025-03-14DOI: 10.1007/s00493-025-00144-4
Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf
{"title":"Sets of r-Graphs that Color All r-Graphs","authors":"Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf","doi":"10.1007/s00493-025-00144-4","DOIUrl":"https://doi.org/10.1007/s00493-025-00144-4","url":null,"abstract":"<p>An <i>r</i>-regular graph is an <i>r</i>-graph, if every odd set of vertices is connected to its complement by at least <i>r</i> edges. Let <i>G</i> and <i>H</i> be <i>r</i>-graphs. An <i>H</i><i>-coloring</i> of <i>G</i> is a mapping <span>(f:E(G) rightarrow E(H))</span> such that each <i>r</i> adjacent edges of <i>G</i> are mapped to <i>r</i> adjacent edges of <i>H</i>. For every <span>(rge 3)</span>, let <span>(mathcal H_r)</span> be an inclusion-wise minimal set of connected <i>r</i>-graphs, such that for every connected <i>r</i>-graph <i>G</i> there is an <span>(H in mathcal H_r)</span> which colors <i>G</i>. The Petersen Coloring Conjecture states that <span>(mathcal H_3)</span> consists of the Petersen graph <i>P</i>. We show that if true, then this is a very exclusive situation. Our main result is that either <span>(mathcal H_3 = {P})</span> or <span>(mathcal H_3)</span> is an infinite set and if <span>(r ge 4)</span>, then <span>(mathcal H_r)</span> is an infinite set. In particular, for all <span>(r ge 3)</span>, <span>(mathcal H_r)</span> is unique. We first characterize <span>(mathcal H_r)</span> and then prove that if <span>(mathcal H_r)</span> contains more than one element, then it is an infinite set. To obtain our main result we show that <span>(mathcal H_r)</span> contains the smallest <i>r</i>-graphs of class 2 and the smallest poorly matchable <i>r</i>-graphs, and we determine the smallest <i>r</i>-graphs of class 2.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"213 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618498","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}
CombinatoricaPub Date : 2025-03-07DOI: 10.1007/s00493-025-00135-5
Oliver Roche-Newton
{"title":"A Lower Bound for the Number of Pinned Angles Determined by a Cartesian Product Set","authors":"Oliver Roche-Newton","doi":"10.1007/s00493-025-00135-5","DOIUrl":"https://doi.org/10.1007/s00493-025-00135-5","url":null,"abstract":"<p>We prove that, for any <span>(B subset {mathbb {R}})</span>, the Cartesian product set <span>(B times B)</span> determines <span>(Omega (|B|^{2+c}))</span> distinct angles.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"53 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143569768","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}
CombinatoricaPub Date : 2025-03-07DOI: 10.1007/s00493-025-00136-4
William Linz
{"title":"L-Systems and the Lovász Number","authors":"William Linz","doi":"10.1007/s00493-025-00136-4","DOIUrl":"https://doi.org/10.1007/s00493-025-00136-4","url":null,"abstract":"<p>Given integers <span>(n> k > 0)</span>, and a set of integers <span>(L subset [0, k-1])</span>, an <i>L</i>-<i>system</i> is a family of sets <span>(mathcal {F}subset left( {begin{array}{c}[n] kend{array}}right) )</span> such that <span>(|F cap F'| in L)</span> for distinct <span>(F, F'in mathcal {F})</span>. <i>L</i>-systems correspond to independent sets in a certain generalized Johnson graph <i>G</i>(<i>n</i>, <i>k</i>, <i>L</i>), so that the maximum size of an <i>L</i>-system is equivalent to finding the independence number of the graph <i>G</i>(<i>n</i>, <i>k</i>, <i>L</i>). The <i>Lovász number</i> <span>(vartheta (G))</span> is a semidefinite programming approximation of the independence number <span>(alpha )</span> of a graph <i>G</i>. In this paper, we determine the leading order term of <span>(vartheta (G(n, k, L)))</span> of any generalized Johnson graph with <i>k</i> and <i>L</i> fixed and <span>(nrightarrow infty )</span>. As an application of this theorem, we give an explicit construction of a graph <i>G</i> on <i>n</i> vertices with a large gap between the Lovász number and the Shannon capacity <i>c</i>(<i>G</i>). Specifically, we prove that for any <span>(epsilon > 0)</span>, for infinitely many <i>n</i> there is a generalized Johnson graph <i>G</i> on <i>n</i> vertices which has ratio <span>(vartheta (G)/c(G) = Omega (n^{1-epsilon }))</span>, which improves on all known constructions. The graph <i>G</i> <i>a fortiori</i> also has ratio <span>(vartheta (G)/alpha (G) = Omega (n^{1-epsilon }))</span>, which greatly improves on the best known explicit construction.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"127 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143570294","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}