A note on Thomassenʼs conjecture

In 1983 C. Thomassen [8] conjectured that for every k , g ∈ N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. A result of Pyber, Szemerédi, and the second author implies that the conjecture is true for every graph G with average d ( G ) ⩾ c k , g l o g Δ ( G ) . engthen this and show that the conjecture holds for every graph G with average d ( G ) ⩾ α ( l o g l o g Δ ( G ) ) β for some constants α, β depending on k and g.