Towards a conjecture on degree conditions for Ramsey goodness of paths

Abstract

Recently, Arag\~{a}o, Marciano, and Mendon\c{c}a [\emph{European J. Combin.}, 2025] conjectured that for any graph $G$ on $n$ vertices satisfying $(r-1)(t-1)k < n \le (r-1)(t-1)(k+1)$, the minimum degree condition $\delta(G) \ge n - \left\lceil \frac{k}{k+1} \left\lceil \frac{n}{r-1} \right\rceil \right\rceil$ guarantees that $G \rightarrow (K_r, P_t)$. In this paper, we prove their conjecture for the regime $k \ge t-3$. Because the parameter $k$ scales linearly with the host graph order $n$, our result establishes the asymptotic truth of the conjecture.