Pub Date : 2025-12-12DOI: 10.1007/s00493-025-00192-w
Karim Alexander Adiprasito, Kaiying Hou, Daishi Kiyohara, Daniel Koizumi, Monroe Stephenson
{"title":"p-Anisotropy on the Moment Curve for Homology Manifolds and Cycles","authors":"Karim Alexander Adiprasito, Kaiying Hou, Daishi Kiyohara, Daniel Koizumi, Monroe Stephenson","doi":"10.1007/s00493-025-00192-w","DOIUrl":"https://doi.org/10.1007/s00493-025-00192-w","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"166 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145753139","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}
Pub Date : 2025-12-03DOI: 10.1007/s00493-025-00183-x
Lucas Aragão, Jonathan Chapman, Miquel Ortega, Victor Souza
The following question was asked by Prendiville: given an r -colouring of the interval $${2, dotsc , N}$${2,⋯,N} , what is the minimum number of monochromatic solutions of the equation $$xy = z$$xy=z ? For $$r=2$$r=2 , we show that there are always asymptotically at least $$(1/2sqrt{2}) N^{1/2} log N$$(1/22)N1/2logN monochromatic solutions, and that the leading constant is sharp. For $$r=3$$r=3 and $$r=4$$r=4 we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations.
Prendiville提出了以下问题：给定区间$${2, dotsc , N}$$ 2,⋯，N{的r着色，方程}$$xy = z$$ x y = z的单色解的最小个数是多少？对于$$r=2$$ r = 2，我们证明了总有至少$$(1/2sqrt{2}) N^{1/2} log N$$ (1 / 2 2) N个1 / 2 log N个单色解，且前导常数是尖锐的。对于$$r=3$$ r = 3和$$r=4$$ r = 4，我们得到紧致的结果，直至一个乘法对数因子。我们还提供了更多颜色和其他乘法方程的边界。
{"title":"On the number of monochromatic solutions to multiplicative equations","authors":"Lucas Aragão, Jonathan Chapman, Miquel Ortega, Victor Souza","doi":"10.1007/s00493-025-00183-x","DOIUrl":"https://doi.org/10.1007/s00493-025-00183-x","url":null,"abstract":"The following question was asked by Prendiville: given an <jats:italic>r</jats:italic> -colouring of the interval <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${2, dotsc , N}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , what is the minimum number of monochromatic solutions of the equation <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$xy = z$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> ? For <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$r=2$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , we show that there are always asymptotically at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(1/2sqrt{2}) N^{1/2} log N$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>log</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> monochromatic solutions, and that the leading constant is sharp. For <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$r=3$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$r=4$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"23 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657518","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}
Pub Date : 2025-12-03DOI: 10.1007/s00493-025-00188-6
Simon Griffiths, Letícia Mattos
We consider the question of how many edge-disjoint near-maximal cliques may be found in the dense Erdős-Rényi random graph G ( n , p ). Recently Acan and Kahn showed that the largest such family contains only $$O(n^2/(log {n})^3)$$O(n2/(logn)3) cliques, with high probability, which disproved a conjecture of Alon and Spencer. We prove the corresponding lower bound, $$Omega (n^2/(log {n})^3)$$Ω(n2/(logn)3) , by considering a random graph process which sequentially selects and deletes near-maximal cliques. To analyse this process we use the Differential Equation Method. We also give a new proof of the upper bound $$O(n^2/(log {n})^3)$$O(n2/(logn)3) and discuss the problem of the precise size of the largest such clique packing.
{"title":"Clique packings in random graphs","authors":"Simon Griffiths, Letícia Mattos","doi":"10.1007/s00493-025-00188-6","DOIUrl":"https://doi.org/10.1007/s00493-025-00188-6","url":null,"abstract":"We consider the question of how many edge-disjoint near-maximal cliques may be found in the dense Erdős-Rényi random graph <jats:italic>G</jats:italic> ( <jats:italic>n</jats:italic> , <jats:italic>p</jats:italic> ). Recently Acan and Kahn showed that the largest such family contains only <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$O(n^2/(log {n})^3)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> cliques, with high probability, which disproved a conjecture of Alon and Spencer. We prove the corresponding lower bound, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Omega (n^2/(log {n})^3)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , by considering a random graph process which sequentially selects and deletes near-maximal cliques. To analyse this process we use the Differential Equation Method. We also give a new proof of the upper bound <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$O(n^2/(log {n})^3)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and discuss the problem of the precise size of the largest such clique packing.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"28 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657519","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}
Pub Date : 2025-11-07DOI: 10.1007/s00493-025-00168-w
Vida Dujmović, Robert Hickingbotham, Jędrzej Hodor, Gwenaël Joret, Hoang La, Piotr Micek, Pat Morin, Clément Rambaud, David R. Wood
We prove that for every planar graph X of treedepth h , there exists a positive integer c such that for every X -minor-free graph G , there exists a graph H of treewidth at most f ( h ) such that G is isomorphic to a subgraph of $$Hboxtimes K_c$$ . This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB, 1986), and treedepth is the optimal parameter in such a result. We give three applications of this result: (1) improved upper bounds for the weak coloring numbers of graphs excluding a given minor, (2) an improved product structure theorem for apex-minor-free graphs, and (3) improved upper bounds for the p -centered chromatic number of graphs excluding a given minor.
{"title":"The Grid-Minor Theorem Revisited","authors":"Vida Dujmović, Robert Hickingbotham, Jędrzej Hodor, Gwenaël Joret, Hoang La, Piotr Micek, Pat Morin, Clément Rambaud, David R. Wood","doi":"10.1007/s00493-025-00168-w","DOIUrl":"https://doi.org/10.1007/s00493-025-00168-w","url":null,"abstract":"We prove that for every planar graph <jats:italic>X</jats:italic> of treedepth <jats:italic>h</jats:italic> , there exists a positive integer <jats:italic>c</jats:italic> such that for every <jats:italic>X</jats:italic> -minor-free graph <jats:italic>G</jats:italic> , there exists a graph <jats:italic>H</jats:italic> of treewidth at most <jats:italic>f</jats:italic> ( <jats:italic>h</jats:italic> ) such that <jats:italic>G</jats:italic> is isomorphic to a subgraph of <jats:inline-formula> <jats:tex-math>$$Hboxtimes K_c$$</jats:tex-math> </jats:inline-formula> . This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB, 1986), and treedepth is the optimal parameter in such a result. We give three applications of this result: (1) improved upper bounds for the weak coloring numbers of graphs excluding a given minor, (2) an improved product structure theorem for apex-minor-free graphs, and (3) improved upper bounds for the <jats:italic>p</jats:italic> -centered chromatic number of graphs excluding a given minor.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"54 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145455351","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}
Pub Date : 2025-11-07DOI: 10.1007/s00493-025-00189-5
Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Paul Wollan
We prove that there is a function f such that every graph with no K -fat $$K_4$$K4 minor is f ( K )-quasi-isometric to a graph with no $$K_4$$K4 minor. This solves the $$K_4$$K4 -case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective $$K_4^-$$K4- -case, which was first established by Fujiwara and Papasoglu.
{"title":"A Characterisation of Graphs Quasi-isometric to $$K_4$$-minor-free Graphs","authors":"Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Paul Wollan","doi":"10.1007/s00493-025-00189-5","DOIUrl":"https://doi.org/10.1007/s00493-025-00189-5","url":null,"abstract":"We prove that there is a function <jats:italic>f</jats:italic> such that every graph with no <jats:italic>K</jats:italic> -fat <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$K_4$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> minor is <jats:italic>f</jats:italic> ( <jats:italic>K</jats:italic> )-quasi-isometric to a graph with no <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$K_4$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> minor. This solves the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$K_4$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> -case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$K_4^-$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> <mml:mo>-</mml:mo> </mml:msubsup> </mml:math> </jats:alternatives> </jats:inline-formula> -case, which was first established by Fujiwara and Papasoglu.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"131 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145455692","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}
Pub Date : 2025-11-07DOI: 10.1007/s00493-025-00186-8
József Balogh, Anton Bernshteyn, Michelle Delcourt, Asaf Ferber, Huy Tuan Pham
A family of r distinct sets $${A_1,ldots , A_r}$${A1,…,Ar} is an r -sunflower if for all $$1 leqslant i < jleqslant r$$1⩽i<j⩽r and $$1 leqslant i' < j'leqslant r$$1⩽i′<j′⩽r , we have $$A_icap A_j = A_{i'}cap A_{j'}$$Ai∩Aj=Ai′∩Aj′ . Erdős and Rado conjectured in 1960 that every family $$mathcal {H}$$H of $$ell $$ℓ -element sets of size at least $$K(r)^ell $$K(r)ℓ contains an r -sunflower, where
一个由r个不同集合组成的族$${A_1,ldots , A_r}$$ A 1，…，A r{是一个r -葵花如果对于所有的}$$1 leqslant i < jleqslant r$$ 1≤i ＆lt； j≤r和$$1 leqslant i' < j'leqslant r$$ 1≤i ' ＆lt; j '≤r，我们有$$A_icap A_j = A_{i'}cap A_{j'}$$ A i∩A j = A i ‘∩A j ’。Erdős和Rado在1960年推测，每个≥$$K(r)^ell $$ K (r) r的族$$mathcal {H}$$ H ($$ell $$) -元素集包含一个r -向日葵，其中K (r)是一个只依赖于r的函数。我们证明了如果$$mathcal {H}$$ H是一个对某些绝对常数$$C > 0$$ C ＆gt； 0和$$|mathcal H| > (C r(log d+log ^*ell ))^ell $$ | H (| ＆gt; (C r (log d + log * r)))) r的不超过d维的$$ell $$ -元素集的族，则$$mathcal {H}$$ H包含一个r -葵花。这改进了Fox、Pach和Suk最近的研究结果。当$$d=1$$ d = 1时，我们得到一个明显的界，即$$|mathcal H| > (r-1)^ell $$ | H | ＆gt; (r - 1) r是充分的。在此过程中，我们建立了有界vc维集合族的Kahn-Kalai猜想的强化，这是一个独立的兴趣。
{"title":"Sunflowers in Set Systems with Small VC-Dimension","authors":"József Balogh, Anton Bernshteyn, Michelle Delcourt, Asaf Ferber, Huy Tuan Pham","doi":"10.1007/s00493-025-00186-8","DOIUrl":"https://doi.org/10.1007/s00493-025-00186-8","url":null,"abstract":"A family of <jats:italic>r</jats:italic> distinct sets <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${A_1,ldots , A_r}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is an <jats:italic>r</jats:italic> -sunflower if for all <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$1 leqslant i < jleqslant r$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>i</mml:mi> <mml:mo><</mml:mo> <mml:mi>j</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$1 leqslant i' < j'leqslant r$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽</mml:mo> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo><</mml:mo> <mml:msup> <mml:mi>j</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>⩽</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , we have <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$A_icap A_j = A_{i'}cap A_{j'}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>j</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . Erdős and Rado conjectured in 1960 that every family <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathcal {H}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>H</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ell $$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -element sets of size at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$K(r)^ell $$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> contains an <jats:italic>r</jats:italic> -sunflower, where ","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"74 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145455690","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}
Pub Date : 2025-11-07DOI: 10.1007/s00493-025-00187-7
Seth Pettie, Gábor Tardos
The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width . The foremost open problem in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $$Pin {0,1}^{ktimes l}$$P∈{0,1}k×l is acyclic , meaning it is the bipartite incidence matrix of a forest, then $$operatorname {Ex}(P,n) = O(nlog ^{C_P} n)$$Ex(P,n)=O(nlogCPn) , where $$operatorname {Ex}(P,n)$$Ex(P,n) is the maximum number of 1s in a P -free $$ntimes n$$n×n 0–1 matrix and $$C_P$$CP is a constant depending only on P . This conjecture has been confirmed on many small patterns, specifically all P with weight at most 5, and all but two with weight 6. The main result of this paper is a clea
禁止0-1矩阵理论推广了Turán-style（二部）子图回避、Davenport-Schinzel理论和zarankiewicz型问题，并在许多领域产生了影响，如离散和计算几何、自调整数据结构的分析以及图参数双宽度的发展。该领域最重要的开放问题是解决2005年的Pach-Tardos猜想，该猜想指出，如果一个禁止模式$$Pin {0,1}^{ktimes l}$$ P∈{0,1}k × 1是无环的，即它是森林的二部关联矩阵，则$$operatorname {Ex}(P,n) = O(nlog ^{C_P} n)$$ Ex (P, n) = O (n log C P n)，其中$$operatorname {Ex}(P,n)$$ Ex (P，n)是无P的$$ntimes n$$ n × n 0-1矩阵中1s的最大个数，$$C_P$$ C P是一个仅与P有关的常数。这个猜想已经在许多小图案上得到了证实，特别是所有P的权重都不超过5，除了两个P的权重都为6。本文的主要结果是对Pach-Tardos猜想的彻底反驳。具体来说，我们证明了$$operatorname {Ex}(S_0,n),operatorname {Ex}(S_1,n) ge n2^{Omega (sqrt{log n})}$$ Ex (S 0, n), Ex (S 1, n)≥n2 Ω （log n），其中$$S_0,S_1$$ S 0， s1是突出的权6模式。我们还证明了整个交替模式（$$(P_t)$$ (P t)）的锐利界，特别是对于每个$$tge 2$$ t≥2,$$operatorname {Ex}(P_t,n)=Theta (n(log n/log log n)^t)$$ Ex (P t, n) = Θ （n (log n / log log n) t）。这是渐近锐界$$omega (nlog n)$$ ω (n log n)的第一个证明。
{"title":"A Refutation of the Pach-Tardos Conjecture for 0–1 Matrices","authors":"Seth Pettie, Gábor Tardos","doi":"10.1007/s00493-025-00187-7","DOIUrl":"https://doi.org/10.1007/s00493-025-00187-7","url":null,"abstract":"The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter <jats:italic>twin width</jats:italic> . The foremost open problem in this area is to resolve the <jats:italic>Pach-Tardos conjecture</jats:italic> from 2005, which states that if a forbidden pattern <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Pin {0,1}^{ktimes l}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>×</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is <jats:italic>acyclic</jats:italic> , meaning it is the bipartite incidence matrix of a forest, then <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$operatorname {Ex}(P,n) = O(nlog ^{C_P} n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>Ex</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mo>log</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:msup> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$operatorname {Ex}(P,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>Ex</mml:mo> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is the maximum number of 1s in a <jats:italic>P</jats:italic> -free <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ntimes n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> 0–1 matrix and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$C_P$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> is a constant depending only on <jats:italic>P</jats:italic> . This conjecture has been confirmed on many small patterns, specifically all <jats:italic>P</jats:italic> with weight at most 5, and all but two with weight 6. The main result of this paper is a clea","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145455693","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}
Pub Date : 2025-11-07DOI: 10.1007/s00493-025-00185-9
Joanna Boyland, William Gasarch, Nathan Hurtig, Robert Rust
{"title":"Big Ramsey Degrees of Countable Ordinals","authors":"Joanna Boyland, William Gasarch, Nathan Hurtig, Robert Rust","doi":"10.1007/s00493-025-00185-9","DOIUrl":"https://doi.org/10.1007/s00493-025-00185-9","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"44 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145455689","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}
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