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Three-color Ramsey number of an odd cycle versus bipartite graphs with small bandwidth

A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a {\em balanced $(β,Δ)$-graph} if it is a bipartite graph with bandwidth at most $β|W|$ and maximum degree at most $Δ$, and it also has a proper 2-coloring $χ:W\rightarrow[2]$ such that $||χ^{-1}(1)|-|χ^{-1}(2)||\leqβ|χ^{-1}(2)|$. In this paper, we prove that for every $γ>0$ and every natural number $Δ$, there exists a constant $β>0$ such that for every balanced $(β,Δ)$-graph $\mathcal{H}$ on $n$ vertices we have $$R(\mathcal{H}, \mathcal{H}, C_n) \leq (3+γ)n$$ for all sufficiently large odd $n$. The upper bound is sharp for several classes of graphs. Let $θ_{n,t}$ be the graph consisting of $t$ internally disjoint paths of length $n$ all sharing the same endpoints. As a corollary, for each fixed $t\geq 1$, $R(θ_{n, t},θ_{n, t}, C_{nt+λ})=(3t+o(1))n,$ where $λ=0$ if $nt$ is odd and $λ=1$ if $nt$ is even. In particular, we have $R(C_{2n},C_{2n}, C_{2n+1})=(6+o(1))n$, which is a special case of a result of Figaj and Łuczak (2018).