All cycle‐complete graph Ramsey numbers r(C_m, K₆)

Abstract

The cycle-complete graph Ramsey number r (C m ,K n ) is the smallest integer N such that every graph G of order N contains a cycle C m on m vertices or has independence number (G) ! n. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that r (C m ,K n ) ¼ (m À 1) (n À 1) þ 1 for all m ! n ! 3 (except r (C 3 ,K 3 ) ¼ 6). This conjecture holds for 3 n 5: In this paper we will present a proof for n ¼ 6 and for all n ! 7 with m ! n 2 À 2n. ß

Large book--cycle Ramsey numbers

“…Bondy and Erdős [6] verified it for n > 3 and ℓ ≥ n 2 − 2, which was slightly improved by Schiermeyer [36] and further by Nikiforov [28] for ℓ ≥ 4n+2. Recently, Keevash, Long and Skokan [24] confirmed this conjecture in a stronger form by proving (5) holds for ℓ ≥ c log n/ log log n, where c > 0 is constant.…”

The Cycle-Complete Graph Ramsey Numbers

“…This has been proved for r = 4 in [14], for r = 5 in [1], and for r = 6 in [12]. In [12] Schiermeyer has also shown that (1) holds for r > 3, p ≥ r 2 − 2r. In this note we prove that (1) holds for all r ≥ 3 and p ≥ 4r + 2.…”

“…Later in [8] Erdős, Faudree, Rousseau, and Schelp conjectured that (1) holds for every p ≥ r ≥ 3, except for p = r = 3. This has been proved for r = 4 in [14], for r = 5 in [1], and for r = 6 in [12]. In [12] Schiermeyer has also shown that (1) holds for r > 3, p ≥ r 2 − 2r.…”

“…We shall use induction on r; for r ≤ 5 (2) follows from the earlier results in [14], [1], and [12]; so we assume that r ≥ 6 and ( 2) holds for all r ′ < r. Assume (2) does not hold for r and p, and let G be a C p+1 -free graph of order rp + 1 with α (G) ≤ r; we shall show that these assumptions lead to a contradiction.…”

All Ramsey $(C_n,K_6)$ critical graphs for large $n$

“…In an attempt to prove Bondy and Erdös conjecture r(C n , K m ) = (n − 1)(m − 1) + 1, for all (n, m) = (3, 3) satisfying n ≥ m ≥ 3 under certain restrictions, Schiermeyer has proved that r(C n , K 6 ) = 5(n − 1) + 1, for n ≥ 6 (see [10,11]). Characterizing all Ramsey critical (C n , K 6 ) graphs boils down to finding all (red/blue) colorings of K r(Cn,K 6 )−1 such that there is no red C n or a blue K 6 .…”