## Abstract

The size Ramsey number of a graph *H* is defined as the minimum number of edges in a graph *G* such that there is a monochromatic copy of *H* in every two-coloring of *E*(*G*). The size Ramsey number was introduced by Erdős, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs (obtained by adding many common neighbors to the vertices of a clique), and starburst graphs (obtained by adding many pendant edges to each vertex of a clique). In this paper, we completely resolve the latter two questions and make substantial progress on the first by determining the size Ramsey number of \(K_{s,t}\) up to a constant factor for all \(t =\Omega (s\log s)\).

This is a preview of subscription content, access via your institution.

## Notes

They only state their result for \(s=t\), but the proof carries through for all \(s \le t\). We present their proof, in this greater generality, in Sect. 2.

They only included the proof for the weaker lower bound \(n^2/2\), but claimed the bound shown. For completeness, we include a proof of the lower bound in Sect. 4.

We are grateful to Mehtaab Sawhney for a suggestion that greatly simplified the proof of this lemma.

Note that, as described, the neighborhood of \(v_j\) in \(I_\ell \) isn’t exactly hypergeometrically distributed, since we first pick a set of “possible” red neighbors hypergeometrically, but then only color red those “possible” neighbors which are truly neighbors in

*G*. Though it doesn’t really matter, this choice makes the analysis slightly simpler.There is also a slight subtlety when \(v_j \in A\), in that, if \(v_j \in I_\ell \) for some interval \(I_\ell \), we only wish to describe the color of the edges from \(v_j\) to the previous vertices \(v_i \in I_\ell \); this is why we add the condition \(i<j\).

This is true if the parts in the pair are disjoint. If not, one can choose a smaller \(\varepsilon \) to guarantee the same end.

## References

Balogh, J., Morris, R., Samotij, W.: Independent sets in hypergraphs. J. Am. Math. Soc.

**28**, 669–709 (2015)Beck, J.: On size Ramsey number of paths, trees, and circuits. I. J. Graph Theory

**7**, 115–129 (1983)Beck, J.: On size Ramsey number of paths, trees and circuits. II. In: Mathematics of Ramsey Theory, Algorithms Combin., vol. 5, pp. 34–45. Springer, Berlin (1990)

Berger, S., Kohayakawa, Y., Maesaka, G.S., Martins, T., Mendonça, W., Mota, G.O., Parczyk, O.: The size-Ramsey number of powers of bounded degree trees. J. Lond. Math. Soc.

**103**, 1314–1332 (2021)Clemens, D., Jenssen, M., Kohayakawa, Y., Morrison, N., Mota, G.O., Reding, D., Roberts, B.: The size-Ramsey number of powers of paths. J. Graph Theory

**91**, 290–299 (2019)Clemens, D., Miralaei, M., Reding, D., Schacht, M., Taraz, A.: On the size-Ramsey number of grid graphs. Comb. Probab. Comput.

**30**, 670–685 (2021)Conlon, D.: The Ramsey number of books, Adv. Comb., Paper No. 3, 12 pp. (2019)

Conlon, D., Fox, J., Sudakov, B.: Recent developments in graph Ramsey theory. In: Surveys in combinatorics 2015, London Math. Soc. Lecture Note Ser., vol. 424, pp. 49–118. Cambridge Univ. Press, Cambridge (2015)

Conlon, D., Fox, J., Wigderson, Y.: Ramsey numbers of books and quasirandomness. Combinatorica

**42**, 309–363 (2022)Conlon, D., Fox, J., Wigderson, Y.: Off-diagonal book Ramsey numbers. Combin. Probab. Comput. 32, 516–545 (2023)

Conlon, D., Gowers, W.T.: Combinatorial theorems in sparse random sets. Ann. Math.

**184**, 367–454 (2016)Conlon, D., Nenadov, R., Trujić, M.: The size-Ramsey number of cubic graphs. Bull. Lond. Math. Soc.

**54**, 2135–2150 (2022)Conlon, D., Nenadov, R., Trujić, M.: On the size-Ramsey number of grids, to appear in Combin. Probab. Comput. Preprint available at arXiv:2202.01654 [math.CO]

Draganić, N., Krivelevich, M., Nenadov, R.: Rolling backwards can move you forward: on embedding problems in sparse expanders. In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 123–134. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2021)

Draganić, N., Krivelevich, M., Nenadov, R.: The size-Ramsey number of short subdivisions. Random Struct. Algorithms

**59**, 68–78 (2021)Erdős, P., Faudree, R.J., Rousseau, C.C., Schelp, R.H.: The size Ramsey number. Period. Math. Hungar.

**9**, 145–161 (1978)Erdős, P., Rousseau, C.C.: The size Ramsey number of a complete bipartite graph. Discrete Math.

**113**, 259–262 (1993)Friedman, J., Pippenger, N.: Expanding graphs contain all small trees. Combinatorica

**7**, 71–76 (1987)Han, J., Jenssen, M., Kohayakawa, Y., Mota, G.O., Roberts, B.: The multicolour size-Ramsey number of powers of paths. J. Comb. Theory Ser. B

**145**, 359–375 (2020)Han, J., Kohayakawa, Y., Letzter, S., Mota, G.O., Parczyk, O.: The size-Ramsey number of 3-uniform tight paths. Adv. Comb., Paper No. 5, 12pp. (2021)

Haxell, P.E., Kohayakawa, Y., Łuczak, T.: The induced size-Ramsey number of cycles. Comb. Probab. Comput.

**4**, 217–239 (1995)Janson, S., Łuczak, T., Rucinski, A.: Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (2000)

Kamčev, N., Liebenau, A., Wood, D.R., Yepremyan, L.: The size Ramsey number of graphs with bounded treewidth. SIAM J. Discret. Math.

**35**, 281–293 (2021)Kohayakawa, Y., Rödl, V., Schacht, M., Szemerédi, E.: Sparse partition universal graphs for graphs of bounded degree. Adv. Math.

**226**, 5041–5065 (2011)Krivelevich, M., Sudakov, B.: Pseudo-random graphs, in More sets, graphs and numbers. In: Bolyai Soc. Math. Stud., vol. 15, pp. 199–262. Springer, Berlin (2006)

Letzter, S., Pokrovskiy, A., Yepremyan, L.: Size-Ramsey numbers of powers of hypergraph trees and long subdivisions. Preprint available at arXiv:2103.01942 [math.CO]

Lovász, L.: Combinatorial Problems and Exercises, 2nd edn. AMS Chelsea Publishing, Providence (2007)

Pikhurko, O.: Asymptotic size Ramsey results for bipartite graphs. SIAM J. Discrete Math.

**16**, 99–113 (2002)Rödl, V., Szemerédi, E.: On size Ramsey numbers of graphs with bounded degree. Combinatorica

**20**, 257–262 (2000)Saxton, D., Thomason, A.: Hypergraph containers. Invent. Math.

**201**, 925–992 (2015)Schacht, M.: Extremal results for random discrete structures. Ann. Math.

**184**, 333–365 (2016)Szabó, T., Vu, V.H.: Turán’s theorem in sparse random graphs. Random Struct. Algorithms

**23**, 225–234 (2003)

## Acknowledgements

We are grateful to Mehtaab Sawhney for suggesting the simplified proof of Lemma 2.3 included here. We would also like to thank the anonymous referees for several helpful comments that improved the presentation of the paper.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

David Conlon’s research was supported by NSF Award DMS-2054452. Jacob Fox’s research was supported by a Packard Fellowship and by NSF Awards DMS-1855635 and DMS-2154169. Yuval Wigderson’s research was supported by NSF GRFP Grant DGE-1656518.

## Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

## About this article

### Cite this article

Conlon, D., Fox, J. & Wigderson, Y. Three Early Problems on Size Ramsey Numbers.
*Combinatorica* **43**, 743–768 (2023). https://doi.org/10.1007/s00493-023-00034-7

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00493-023-00034-7