Journal of Combinatorial Theory Series B最新文献

{"title":"Rigidity and reconstruction in matroids of highly connected graphs","authors":"Dániel Garamvölgyi","doi":"10.1016/j.jctb.2026.01.003","DOIUrl":"10.1016/j.jctb.2026.01.003","url":null,"abstract":"<div><div>A <em>graph matroid family</em> <span><math><mi>M</mi></math></span> is a family of matroids <span><math><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> defined on the edge set of each finite graph <em>G</em> in a compatible and isomorphism-invariant way. We say that <span><math><mi>M</mi></math></span> has the <em>Whitney property</em> if there is a constant <em>c</em> such that every <em>c</em>-connected graph <em>G</em> is uniquely determined by <span><math><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Similarly, <span><math><mi>M</mi></math></span> has the <em>Lovász-Yemini property</em> if there is a constant <em>c</em> such that for every <em>c</em>-connected graph <em>G</em>, <span><math><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has maximal rank among graphs on the same number of vertices.</div><div>We show that if <span><math><mi>M</mi></math></span> is unbounded (that is, there is no absolute constant bounding the rank of <span><math><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for every <em>G</em>), then <span><math><mi>M</mi></math></span> has the Whitney property if and only if it has the Lovász-Yemini property. We also give a complete characterization of these properties in the bounded case. As an application, we show that if some graph matroid families have the Whitney property, then so does their union. Finally, we show that every 1-extendable graph matroid family has the Lovász-Yemini (and thus the Whitney) property. These results unify and extend a number of earlier results about graph reconstruction from an underlying matroid.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 211-244"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Diameter bounds for distance-regular graphs via long-scale Ollivier Ricci curvature","authors":"Kaizhe Chen ,&nbsp;Shiping Liu","doi":"10.1016/j.jctb.2025.12.002","DOIUrl":"10.1016/j.jctb.2025.12.002","url":null,"abstract":"<div><div>In this paper, we derive new sharp diameter bounds for distance regular graphs, which better answer a problem raised by Neumaier and Penjić in many cases. Our proof is built upon a relation between the diameter and long-scale Ollivier Ricci curvature of a graph, which can be considered as an improvement of the discrete Bonnet-Myers theorem. Our method further leads to significant improvements of existing diameter bounds for amply regular graphs and <span><math><mo>(</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span>-graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 104-117"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145732369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Corrigendum to “On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes” [J. Comb. Theory, Ser. B 103 (4) (2013) 428–467]","authors":"Victor Chepoi, Mark Hagen","doi":"10.1016/j.jctb.2026.04.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2026.04.001","url":null,"abstract":"","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"19 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147736268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"3-colorable planar graphs have an intersection segment representation using 3 slopes","authors":"Daniel Gonçalves","doi":"10.1016/j.jctb.2025.11.005","DOIUrl":"10.1016/j.jctb.2025.11.005","url":null,"abstract":"<div><div>In his PhD Thesis E.R. Scheinerman conjectured that planar graphs are intersection graphs of line segments in the plane. This conjecture was proved with two different approaches by J. Chalopin and the author, and by the author, L. Isenmann, and C. Pennarun. In the case of 3-colorable planar graphs E.R. Scheinerman conjectured that it is possible to restrict the set of slopes used by the segments to only 3 slopes. Here we prove this conjecture by using an approach introduced by S. Felsner to deal with contact representations of planar graphs with homothetic triangles.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 234-256"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145611785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On digraphs without onion star immersions","authors":"Łukasz Bożyk ,&nbsp;Oscar Defrain ,&nbsp;Karolina Okrasa ,&nbsp;Michał Pilipczuk","doi":"10.1016/j.jctb.2025.11.009","DOIUrl":"10.1016/j.jctb.2025.11.009","url":null,"abstract":"<div><div>The <em>t-onion star</em> is the digraph obtained from a star with 2<em>t</em> leaves by replacing every edge by a triple of arcs, where in <em>t</em> triples we orient two arcs away from the center, and in the remaining <em>t</em> triples we orient two arcs towards the center. Note that the <em>t</em>-onion star contains, as an immersion, every digraph on <em>t</em> vertices where each vertex has outdegree at most 2 and indegree at most 1, or vice versa.</div><div>We investigate the structure in digraphs that exclude a fixed onion star as an immersion. The main discovery is that in such digraphs, for some duality statements true in the undirected setting we can prove their directed analogues. More specifically, we show the next two statements.<ul><li><span>•</span><span><div>There is a function <span><math><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> satisfying the following: If a digraph <em>D</em> contains a set <em>X</em> of <span><math><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn></math></span> vertices such that for any <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></math></span> there are <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> arc-disjoint paths from <em>x</em> to <em>y</em>, then <em>D</em> contains the <em>t</em>-onion star as an immersion.</div></span></li><li><span>•</span><span><div>There is a function <span><math><mi>g</mi><mo>:</mo><mi>N</mi><mo>×</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> satisfying the following: If <em>x</em> and <em>y</em> is a pair of vertices in a digraph <em>D</em> such that there are at least <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> arc-disjoint paths from <em>x</em> to <em>y</em> and there are at least <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> arc-disjoint paths from <em>y</em> to <em>x</em>, then either <em>D</em> contains the <em>t</em>-onion star as an immersion, or there is a family of 2<em>k</em> pairwise arc-disjoint paths with <em>k</em> paths from <em>x</em> to <em>y</em> and <em>k</em> paths from <em>y</em> to <em>x</em>.</div></span></li></ul></div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 257-272"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145689360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Maker-Breaker percolation game on the square lattice","authors":"Vojtěch Dvořák,&nbsp;Adva Mond,&nbsp;Victor Souza","doi":"10.1016/j.jctb.2025.10.010","DOIUrl":"10.1016/j.jctb.2025.10.010","url":null,"abstract":"<div><div>We study the <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span> Maker-Breaker percolation game on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, introduced by Day and Falgas-Ravry. In this game, on each of their turns, Maker and Breaker claim respectively <em>m</em> and <em>b</em> unclaimed edges of the square lattice <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Breaker wins if the component containing the origin becomes finite when his edges are deleted from <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Maker wins if she can indefinitely avoid a win of Breaker. We show that Breaker has a winning strategy for the <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span> game whenever <span><math><mi>b</mi><mo>⩾</mo><mo>(</mo><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mn>14</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>m</mi></math></span>, breaking the ratio 2 barrier proved by Day and Falgas-Ravry.</div><div>Addressing further questions of Day and Falgas-Ravry, we show that Breaker can win the <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mn>2</mn><mi>m</mi><mo>)</mo></math></span> game even if he allows Maker to claim <em>c</em> edges before the game starts, for any integer <em>c</em>, and that he can moreover win rather quickly as a function of <em>c</em>.</div><div>We also consider the game played on the so-called polluted board, obtained after performing Bernoulli bond percolation on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with parameter <em>p</em>. We show that for the <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> game on the polluted board, Breaker almost surely has a winning strategy whenever <span><math><mi>p</mi><mo>⩽</mo><mn>0.6298</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 31-66"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145499108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bisection width, discrepancy, and eigenvalues of hypergraphs","authors":"Eero Räty ,&nbsp;István Tomon","doi":"10.1016/j.jctb.2025.11.003","DOIUrl":"10.1016/j.jctb.2025.11.003","url":null,"abstract":"<div><div>A celebrated result of Alon from 1993 states that any <em>d</em>-regular graph on <em>n</em> vertices (where <span><math><mi>d</mi><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>9</mn></mrow></msup><mo>)</mo></math></span>) has a bisection with at most <span><math><mfrac><mrow><mi>d</mi><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>Ω</mi><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mi>d</mi></mrow></msqrt></mrow></mfrac><mo>)</mo><mo>)</mo></math></span> edges, and this is optimal. Recently, this result was greatly extended by Räty, Sudakov, and Tomon. We build on the ideas of the latter, and use a semidefinite programming inspired approach to prove the following variant for hypergraphs: every <em>r</em>-uniform <em>d</em>-regular hypergraph on <em>n</em> vertices (where <span><math><mi>d</mi><mo>≪</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>) has a bisection of size at most<span><span><span><math><mfrac><mrow><mi>d</mi><mi>n</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>c</mi></mrow><mrow><msqrt><mrow><mi>d</mi></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mo>,</mo></math></span></span></span> for some <span><math><mi>c</mi><mo>=</mo><mi>c</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>&gt;</mo><mn>0</mn></math></span>. This bound is the best possible up to the precise value of <em>c</em>. Moreover, a bisection achieving this bound can be found by a polynomial-time randomized algorithm.</div><div>The minimum bisection is closely related to discrepancy. We also prove sharp bounds on the discrepancy and so called positive discrepancy of hypergraphs, extending results of Bollobás and Scott. Furthermore, we discuss implications about Alon-Boppana type bounds. We show that if <em>H</em> is an <em>r</em>-uniform <em>d</em>-regular hypergraph, then certain notions of second largest eigenvalue <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> associated with the adjacency tensor satisfy <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math></span>, improving results of Li and Mohar.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 186-215"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145559890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cutting corners","authors":"Andrey Kupavskii ,&nbsp;Arsenii Sagdeev ,&nbsp;Dmitrii Zakharov","doi":"10.1016/j.jctb.2025.11.008","DOIUrl":"10.1016/j.jctb.2025.11.008","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We say that a subset &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is &lt;em&gt;exponentially Ramsey&lt;/em&gt; if there exists &lt;span&gt;&lt;math&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; stands for the minimum number of colors in a coloring of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; such that no copy of &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is monochromatic. One important result in Euclidean Ramsey theory is due to Frankl and Rödl, and states the following (under some mild extra conditions): if both &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are exponentially Ramsey then so is their Cartesian product. Applied several times to simple two-point sets &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, this result implies that any subset &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of a ‘hyperrectangle’ &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is exponentially Ramsey.&lt;/div&gt;&lt;div&gt;However, generally, such ‘embeddings’ of &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; result in very inefficient bounds on the aforementioned &lt;em&gt;ε&lt;/em&gt;. In this paper, we present another way of combining exponentially Ramsey sets, which gives much better estimates in some important cases. In particular, we show that the chromatic number of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with a forbidden equilateral triangle satisfies &lt;span&gt;&lt;math&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;△&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1.0742&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, greatly improving upon the previous constant 1.0144. We also obtain similar strong results for regular simplices of larger dimensions, as well as for related geometric Ramsey-type questions in Manhattan norm.&lt;/div&gt;&lt;div&gt;We then show that the same technique implies several interesting corollaries in other combi","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 273-292"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145690151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Intersecting families with covering number 3","authors":"Andrey Kupavskii","doi":"10.1016/j.jctb.2025.11.004","DOIUrl":"10.1016/j.jctb.2025.11.004","url":null,"abstract":"<div><div>The covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> the largest intersecting family of <em>k</em>-element subsets of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> with covering number 3. In this paper, we essentially settle this problem, showing that the same family is extremal for any <span><math><mi>k</mi><mo>≥</mo><mn>100</mn></math></span> and <span><math><mi>n</mi><mo>&gt;</mo><mn>2</mn><mi>k</mi></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 216-233"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145583813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reconfiguration of basis pairs in regular matroids","authors":"Kristóf Bérczi ,&nbsp;Bence Mátravölgyi ,&nbsp;Tamás Schwarcz","doi":"10.1016/j.jctb.2025.10.009","DOIUrl":"10.1016/j.jctb.2025.10.009","url":null,"abstract":"<div><div>In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns the exchange distance of two matroid basis sequences, a problem that appears in several areas of computer science and mathematics. White (1980) proposed a conjecture for the characterization of two basis sequences being reachable from each other by symmetric exchanges, which received a significant interest also in algebra due to its connection to toric ideals and Gröbner bases. In this work, we verify White's conjecture for basis sequences of length two in regular matroids, a problem that was formulated as a separate question by Farber, Richter, and Shank (1985) and Andres, Hochstättler, and Merkel (2014). Most of previous work on White's conjecture has not considered the question from an algorithmic perspective. We study the problem from an optimization point of view: our proof implies a polynomial algorithm for determining a sequence of symmetric exchanges that transforms a basis pair into another, thus providing the first polynomial upper bound on the exchange distance of basis pairs in regular matroids. As a byproduct, we verify a conjecture of Gabow (1976) on the serial symmetric exchange property of matroids for the regular case.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 105-142"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145529317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}