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}
Pub Date : 2025-10-10DOI: 10.1007/s00493-025-00171-1
Miloš Trujić
In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph <inline-formula><alternatives><mml:math><mml:msub><mml:mi>G</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G_{n, p}$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_171_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula> for <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mo>⩾</mml:mo><mml:mi>C</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>log</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$p geqslant C(log n/n)^{1/Delta }$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_171_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph <italic>H</italic> with <inline-formula><alternatives><mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>⩽</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta (H) leqslant Delta$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_171_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula>. However, this is typically only optimal when <inline-formula><alternatives><mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} u
In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph Gn,pdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G_{n, p}$$end{document} for p⩾C(logn/n)1/Δdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$p geqslant C(log n/n)^{1/Delta }$$end{document}, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph H with Δ(H)⩽Δdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta (H) leqslant Delta$$end{document}. However, this is typically only optimal when Δ∈{2,3}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta in {2,3}$$end{document} and H either contains a triangle (Δ=2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta = 2$$end{document}) or many copies of K4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$K_4$$end{document} (Δ=3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta = 3$$end{document}). We go beyond this barrier for the first time and present a sparse blow-up lemma for cycles C2k-1,C2kdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$C_{2k-1}, C_{2k}$$end{document}, for all k⩾2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k geqslant 2$$end{document}, and densities p⩾Cn-(k-1)/kdocumentclass[12pt]{minimal} usepackage{amsmath}
{"title":"Blow-up Lemma for Cycles in Sparse Random Graphs","authors":"Miloš Trujić","doi":"10.1007/s00493-025-00171-1","DOIUrl":"https://doi.org/10.1007/s00493-025-00171-1","url":null,"abstract":"In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph <inline-formula><alternatives><mml:math><mml:msub><mml:mi>G</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G_{n, p}$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_171_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula> for <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mo>⩾</mml:mo><mml:mi>C</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>log</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$p geqslant C(log n/n)^{1/Delta }$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_171_Article_IEq2.gif\"></inline-graphic></alternatives></inline-formula>, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph <italic>H</italic> with <inline-formula><alternatives><mml:math><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⩽</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta (H) leqslant Delta$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_171_Article_IEq3.gif\"></inline-graphic></alternatives></inline-formula>. However, this is typically only optimal when <inline-formula><alternatives><mml:math><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} u","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"29 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914896","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 : 2025-10-10DOI: 10.1007/s00493-025-00175-x
Yunsong Gan, Pablo Spiga, Binzhou Xia
This paper represents a significant leap forward in the problem of enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of Cayley (di)graphs show that almost all finite Cayley (di)graphs have the smallest possible automorphism group. Extending the scope of these results, we enumerate (di)graphs admitting a fixed semiregular group of automorphisms with m orbits. Moreover, we consider the more intricate inquiry of prohibiting arcs within each orbit, where the special case m=2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$m=2$$end{document} is known as the problem of finding Haar graphical representations (HGRs). We significantly advance the understanding of HGRs by proving that the proportion of HGRs among Haar graphs of a finite nonabelian group approaches 1 as the group order grows. As a corollary, we obtain an improved bound on the proportion of DRRs among Cayley digraphs in the solution of Morris and the second author to the Babai-Godsil conjecture.
This paper represents a significant leap forward in the problem of enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of Cayley (di)graphs show that almost all finite Cayley (di)graphs have the smallest possible automorphism group. Extending the scope of these results, we enumerate (di)graphs admitting a fixed semiregular group of automorphisms with m orbits. Moreover, we consider the more intricate inquiry of prohibiting arcs within each orbit, where the special case m=2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$m=2$$end{document} is known as the problem of finding Haar graphical representations (HGRs). We significantly advance the understanding of HGRs by proving that the proportion of HGRs among Haar graphs of a finite nonabelian group approaches 1 as the group order grows. As a corollary, we obtain an improved bound on the proportion of DRRs among Cayley digraphs in the solution of Morris and the second author to the Babai-Godsil conjecture.
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