Size‐Ramsey Numbers of Structurally Sparse Graphs.
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| Title: | Size‐Ramsey Numbers of Structurally Sparse Graphs. |
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| 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] |
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| Database: | Engineering Source |
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