In the linked paper by Vaclav Chvatal and Frank Harary, they conjecture that the Ramsey number of two general graphs $F_1,F_2$ satisfies
$$r(F_1,F_2) \ge \min\left( r(F_1,F_1), r(F_2,F_2) \right)$$
This paper is really old, im wondering if someone knows if this conjecture has been (dis)proven (or any progress on it)?
(PS I'd really appreciate if someone could comment below any other references about generalised ramsey numbers. For example, I already saw Chvatal's proof of the $r(T,K_n)$ an the proofs of $r(Q_n,K_t)$)
The conjecture is false, since $r(P_5,K_{1,3})=5$ while $r(P_5,P_5)=r(K_{1,3},K_{1,3})=6$; for details see my answer to the old question Generalized Ramsey Numbers, or see pp. 43-44 of Michael Capobianco and John C. Molluzzo, Examples and Counterexamples in Graph Theory, North-Holland, New York, 1978, where this counterexsmple is attributed to Galvin. I suspect the original reference is this paper which I haven't seen: Frank Harary, Recent results on generalized Ramsey theory for graphs, in Graph Theory and Applications (Y. Alavi, D. Lick, and A. White, eds.), Springer, New York, 1972, pp. 125-138.