Towards a conjecture on degree conditions for Ramsey goodness of paths

TL;DR

This paper proves a conjecture relating minimum degree conditions to Ramsey properties involving paths and cliques in large graphs, confirming the conjecture asymptotically for certain parameter ranges.

Contribution

It establishes the conjecture for the case where parameter k is at least t-3, confirming its asymptotic validity in that regime.

Findings

Proves the conjecture for k ≥ t-3.

Confirms the asymptotic truth of the conjecture for large graphs.

Provides a new understanding of 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.