Path-kipas Ramsey numbers

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