Three (two)-color Ramsey numbers for paths, matchings and the pendant triangle

For a given positive integer k and given graphs G 1 , … , G k , the k -color Ramsey number R ( G 1 , … , G k ) is defined as the smallest integer n such that every k -edge-coloring of a complete graph of order n contains a monochromatic copy of G i in color i for some i ∈ { 1 , … , k } . In this paper, we consider R ( P t , S 3 + ) , R ( m K 2 , S 3 + ) and R ( P t , m K 2 , S 3 + ) , where P t is a path of order t , m K 2 is a matching of size m , and S 3 + is the pendant triangle obtained from a triangle by adding a pendant edge to a vertex of the triangle. Specifically, we prove that R ( P t , S 3 + ) = 2 t − 1 for each integer t ≥ 3 and R ( m K 2 , S 3 + ) = 2 m + 1 for each integer m ≥ 2 . We further prove that R ( P 3 , m K 2 , S 3 + ) = 2 m + 2 and R ( P 4 , m K 2 , S 3 + ) = 2 m + 4 for each integer m ≥ 2 . Moreover, we prove that 2 m + t ≤ R ( P t , m K 2 , S 3 + ) ≤ 2 m + 2 t − 3 for each pair of integers t ≥ 5 and m ≥ 2 .