Abstract. A graph is [Formula: see text]-Ramsey if any red-blue coloring of its edges contains either a red copy of [Formula: see text] or a blue copy of [Formula: see text]. The size Ramsey number is the minimum number of edges contained in a [Formula: see text]-Ramsey graph. Generalizing the notion of size Ramsey numbers, the [Formula: see text]-Ramsey number [Formula: see text] is defined to be the minimum number of copies of [Formula: see text] in a [Formula: see text]-Ramsey graph. It is easy to see that [Formula: see text]. Recently, Fox, Tidor, and Zhang showed that equality holds in this bound when [Formula: see text] and [Formula: see text], i.e., [Formula: see text]. They further conjectured that [Formula: see text] for all [Formula: see text], in response to a question of Spiro. In this work, we study the off-diagonal variant of this conjecture: is it true that [Formula: see text] whenever [Formula: see text]? Harnessing the constructions used in the recent breakthrough work of Mattheus and Verstraëte on the asymptotics of [Formula: see text], we show that when [Formula: see text] is 3 or 4, the above equality holds up to a lower order term in the exponent.

Abstract. We prove that if [Formula: see text] is a subgraph of a complete multipartite graph [Formula: see text], then [Formula: see text] contains a connected component [Formula: see text] satisfying [Formula: see text]. We use this to prove that every 3-coloring of the edges of a complete graph contains a monochromatic connected subgraph with at least [Formula: see text] of the edges. We further show that such a coloring has a monochromatic circuit with a fraction [Formula: see text] of the edges. This verifies a conjecture of Conlon and Tyomkyn. Moreover, for general [Formula: see text], we show that every [Formula: see text]-coloring of the edges of [Formula: see text] contains a monochromatic connected subgraph with at least [Formula: see text] edges.