The Ramsey number of mixed-parity cycles I

الأصل · EN

Denote by R(G₁, G₂, G₃) the minimum integer N such that any three-colouring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph Gᵢ coloured with colour i for some i∈1,2,3. In a series of three papers of which this is the first, we consider the case where G₁, G₂ and G₃ are cycles of mixed parity. Specifically, in this and the subsequent paper, we consider R(Cₙ,Cₘ,Cℓ), where n and m are even and ℓ is odd. Figaj and Łuczak determined an asymptotic result for this case, which we improve upon to give an exact result. We prove that for n,m and ℓ sufficiently large R(Cₙ,Cₘ,Cℓ)={2n+m-3, n+2m-3, 12 n +12 m + ℓ - 2}. In the case that the longest cycle is of even length, the proof in this paper is self-contained. However, in the case that the longest cycle is of odd length, we require an additional technical result, the proof of which makes up the majority of the subsequent paper.