Ramsey Numbers Involving Graphs with Long Suspended Paths

Generalizations of a Ramsey‐theoretic result of chvátal

“…In addition, we briefly mention some work which is closely related to k-goodness . Theorem 2 .1 [4] . Let k be fixed and let G be a connected graph.…”

“…Actually, the theorem proved in [4] replaces r(Kk, G) by r(F, G), where F is an arbitrary graph . The general theorem involves a notion of "Fgoodness," a concept whose definition is slightly technical, so we mention here only the special case represented by Theorem 2 .1 .…”

Ramsey numbers for certain k-graphs

“…It may be possible, however, to obtain the values for trees containing sufficiently long "suspended" A>paths (see Burr (1979) and Burr and Erdos (1979)). Further progress could no doubt be made on various specific cases, but we have found even the precise determination of r(S(9, 3), C(4, 3)) to be surprisingly difficult.…”

Ramsey numbers for the pair sparse graph-path or cycle

“…Let G and 77 be simple graphs. The Ramsey number r(G, H) is the smallest integer « such that for each graph F on « vertices, either G is a subgraph of F or 77 is a subgraph of F, the complement of F. Calculation of r(G, H) for particular pairs of graphs G and 77 has received considerable attention, and a survey of such results can be found in [2].…”

“…Let G and 77 be simple graphs. The Ramsey number r(G, H) is the smallest integer « such that for each graph F on « vertices, either G is a subgraph of F or 77 is a subgraph of F, the complement of F. Calculation of r(G, H) for particular pairs of graphs G and 77 has received considerable attention, and a survey of such results can be found in [2].Chvátal [5] proved that if Tn is a tree on « vertices and Km is a complete graph on m vertices, then r(Tn, Km) = (n -l)(m -1) + 1. In [4] it was shown that if Tn is replaced by a sparse connected graph Gn on « vertices the Ramsey number remains the same (i.e.…”