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Size‐Ramsey Numbers of Structurally Sparse Graphs.

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Title: Size‐Ramsey Numbers of Structurally Sparse Graphs.
Authors: Draganić, Nemanja1 (AUTHOR), Kaufmann, Marc2 (AUTHOR), Correia, David Munhá1 (AUTHOR), Petrova, Kalina2 (AUTHOR), Steiner, Raphael2 (AUTHOR) raphaelmario.steiner@inf.ethz.ch
Source: Random Structures & Algorithms. Mar2026, Vol. 68 Issue 2, p1-18. 18p.
Subjects: Ramsey numbers, Sparse graphs, Graph theory, Planar graphs, Ramsey theory
Abstract: Size‐Ramsey numbers are a central notion in combinatorics and have been widely studied since their introduction by Erdős, Faudree, Rousseau, and Schelp in 1978. Research has mainly focused on the size‐Ramsey numbers of n$$ n $$‐vertex graphs with constant maximum degree Δ$$ \Delta $$. For example, graphs which also have constant treewidth are known to have linear size‐Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from O(n3/2)$$ O\left({n}^{3/2}\right) $$ to O(n5/4)$$ O\left({n}^{5/4}\right) $$ by Conlon, Nenadov and Trujić. In this paper, we study a common generalization of these problems and establish new bounds on the size‐Ramsey numbers in terms of treewidth (which may grow as a function of n$$ n $$). As a special case, this yields a bound of Õ(n3/2−1/2Δ)$$ \overset{\widetilde }{O}\left({n}^{3/2-1/2\Delta}\right) $$ for proper minor‐closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Kamcev, Liebenau, Wood and Yepremyan. Our proof combines methods from structural graph theory and classic Ramsey‐theoretic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth. [ABSTRACT FROM AUTHOR]
Copyright of Random Structures & Algorithms is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Size‐Ramsey Numbers of Structurally Sparse Graphs.
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  Data: <searchLink fieldCode="AR" term="%22Draganić%2C+Nemanja%22">Draganić, Nemanja</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kaufmann%2C+Marc%22">Kaufmann, Marc</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Correia%2C+David+Munhá%22">Correia, David Munhá</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Petrova%2C+Kalina%22">Petrova, Kalina</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Steiner%2C+Raphael%22">Steiner, Raphael</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> raphaelmario.steiner@inf.ethz.ch</i>
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  Data: <searchLink fieldCode="JN" term="%22Random+Structures+%26+Algorithms%22">Random Structures & Algorithms</searchLink>. Mar2026, Vol. 68 Issue 2, p1-18. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Ramsey+numbers%22">Ramsey numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Sparse+graphs%22">Sparse graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Graph+theory%22">Graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Planar+graphs%22">Planar graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Ramsey+theory%22">Ramsey theory</searchLink>
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  Data: Size‐Ramsey numbers are a central notion in combinatorics and have been widely studied since their introduction by Erdős, Faudree, Rousseau, and Schelp in 1978. Research has mainly focused on the size‐Ramsey numbers of n$$ n $$‐vertex graphs with constant maximum degree Δ$$ \Delta $$. For example, graphs which also have constant treewidth are known to have linear size‐Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from O(n3/2)$$ O\left({n}^{3/2}\right) $$ to O(n5/4)$$ O\left({n}^{5/4}\right) $$ by Conlon, Nenadov and Trujić. In this paper, we study a common generalization of these problems and establish new bounds on the size‐Ramsey numbers in terms of treewidth (which may grow as a function of n$$ n $$). As a special case, this yields a bound of Õ(n3/2−1/2Δ)$$ \overset{\widetilde }{O}\left({n}^{3/2-1/2\Delta}\right) $$ for proper minor‐closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Kamcev, Liebenau, Wood and Yepremyan. Our proof combines methods from structural graph theory and classic Ramsey‐theoretic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of Random Structures & Algorithms is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1002/rsa.70059
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      – Code: eng
        Text: English
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        Type: general
      – SubjectFull: Sparse graphs
        Type: general
      – SubjectFull: Graph theory
        Type: general
      – SubjectFull: Planar graphs
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      – SubjectFull: Ramsey theory
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      – TitleFull: Size‐Ramsey Numbers of Structurally Sparse Graphs.
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              M: 03
              Text: Mar2026
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              Y: 2026
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