The Ramsey Numbers of Paths Versus Kipases

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 ( P n , K ˆ m ) , where P n is a path on n vertices and K ˆ m is the graph obtained from the join of K 1 and P m . We determine the exact values of R ( P n , K ˆ m ) for the following values of n and m: 1 ⩽ n ⩽ 5 and m ⩾ 3 ; n ⩾ 6 and (m is odd, 3 ⩽ m ⩽ 2 n − 1 ) or (m is even, 4 ⩽ m ⩽ n + 1 ); n = 6 or 7 and m = 2 n − 2 or m ⩾ 2 n ; n ⩾ 8 and m = 2 n − 2 or m = 2 n or ( q ⋅ n − 2 q + 1 ⩽ m ⩽ q ⋅ n − q + 2 with 3 ⩽ q ⩽ n − 5 ) or m ⩾ ( n − 3 ) 2 ; odd n ⩾ 9 and ( q ⋅ n − 3 q + 1 ⩽ m ⩽ q ⋅ n − 2 q with 3 ⩽ q ⩽ ( n − 3 ) / 2 ) or ( q ⋅ n − q − n + 4 ⩽ m ⩽ q ⋅ n − 2 q with ( n − 1 ) / 2 ⩽ q ⩽ n − 4 ).