The path–linear-forest quasar Ramsey number conjecture

Let $P_n$ be the path on $n$ vertices, let $F$ be a non-empty linear forest on $m$ vertices, let $K_1\vee F$ denote the join of $F$ with a single vertex, and let $o(F)$ be the number of odd-order components of $F$. Path–linear-forest quasar Ramsey conjecture. If

$$n\geq 2\quad\text{and}\quad n+1\leq m\leq 2n-1,$$

then

$$R(P_n,K_1\vee F)=\max\left\{2n-1,\left\lceil\frac{3m}{2}\right\rceil-1, m+n-o(F)-2\right\}.$$

This would determine the remaining Ramsey numbers of paths versus proper quasars of the form $K_1\vee F$ in the range $n+1\leq m\leq 2n-1$; the preceding results establish matching bounds when every component of $F$ has even order, but the general case is left open.