On-line Ramsey Numbers of Paths and Cycles

Abstract

Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of G or a blue copy of H for some fixed graphs G and H. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey numberr(G, H). In this paper, we consider the case where G is a path P k . We prove thatr(P 3 , P ℓ+1 ) = ⌈5ℓ/4⌉ =r(P 3 , C ℓ ) for all ℓ ≥ 5, and determiner(P 4 , P ℓ+1 ) up to an additive constant for all ℓ ≥ 3. We also prove some general lower bounds for on-line Ramsey numbers of the form r(P k+1 , H).

On Path Convex Ramsey Numbers

“…Also the Ramsey numbers for the pair sparse graph‐path or cycle were found by S.A. Burr, P. Erdős, R.J. Faudree, C.C. Rousseau, and R.H. Schelp (see also ).…”

A Note on On-Line Ramsey Numbers for Some Paths

“…The terminology, definitions and some descriptions are taken from two previous works by the first author, namely [1,2].…”

“…A significant amount of effort has been focused on the special case where G is a path P 3 . Cyman, Dzido, Lapinskas and Lo [1] have determined r(P 3 , P +1 ) andr(P 3 , C ) exactly (where P s is a path on s vertices).…”

Restricted size Ramsey number for P3 versus cycle

“…Therefore, the size of K r is the upper bound for the restricted size Ramsey number of G and H and the restricted size Ramsey number must be greater or equal to the size Ramsey number for a given pair of graphs. In addition, we haver(G, H) ≤r(G, H), wherer(G, H) is the on-line Ramsey number (the definition and properties of these numbers can be found in [2]). If both G and H are complete graphs then F = K r (see [3]).…”