Path-fan Ramsey numbers

For two given graphs G and H , the Ramsey number R(G, H) is the smallest positive integer p such that for every graph F on p vertices the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we study the Ramsey numbers R(Pn, Fm), where Pn is a path on n vertices and Fm is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (Fm is the join of K1 and mK2). We determine the exact values of R(Pn, Fm) for the following values of n and m: 1 ≤ n ≤ 5 and m ≥ 2; n ≥ 6 and 2 ≤ m ≤ (n + 1)/2; n = 6 or 7 and m ≥ n− 1; n ≥ 8 and m = n− 1 or m = n or (q · n− 2q + 1)/2 ≤ m ≤ (q · n− q + 2)/2 with 3 ≤ q ≤ n− 5 or m ≥ (n− 3)/2; odd n ≥ 9 and ((q · n − 3q + 1)/2 ≤ m ≤ (q · n − 2q)/2 with 3 ≤ q ≤ (n − 3)/2) or ((q · n− q −n + 4)/2 ≤ m ≤ (q ·n− 2q)/2 with (n− 1)/2 ≤ q ≤ n− 5). Besides that, we give nontrivial lower bounds and upper bounds for R(Pn, Fm) for the other values of m and n.