Treffer: 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, Km), where Pn is a path on n vertices and Km is the graph obtained from the join of Kl and Pm. We determine the exact values of R(Pn, Km) 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)2; 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, Km) for the other values of m and n.