We prove that for any two graphs G and H , the edges of G can be strongly separated by a collection of linearly many subdivisions of H and single edges. This confirms a conjecture of Botler and Naia.

Abstract. We study conditions under which an edge-colored hypergraph has a particular substructure that contains more than the trivially guaranteed number of monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions. This answers recent questions of Gishboliner, Glock, and Sgueglia and of Balogh, Treglown, and Zárate-Guerén.

We propose the following conjecture: For every fixed $\alpha\in [0,\frac 13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph. Our main result is an approximate version of the conjecture for bounded degree trees and large dense host graphs. We also show that our conjecture is asymptotically best possible. The proof of the approximate result relies on a second result, which we believe to be interesting on its own. Namely, we can embed any bounded degree tree into host graphs of minimum/maximum degree asymptotically exceeding $\frac k2$ and $\frac 43k$, respectively, as long as the host graph avoids a specific structure.

We conjecture that every graph of minimum degree at least $\frac k2$ and maximum degree at least 2 k contains all trees with k edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our result relies on a general embedding tool for embedding trees into graphs of certain structure. This tool also has implications for the Erdös--Sós conjecture and the $\frac 23$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.