# Three Early Problems on Size Ramsey Numbers

## 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 $$t =\Omega (s\log s)$$.

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1. 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.

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.

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

4. 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 $$i<j$$.

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

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## 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.

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Correspondence to Yuval Wigderson.

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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.

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