Multicolour Ramsey Numbers of Odd Cycles

Monochromatic tree covers and Ramsey numbers for set-coloured graphs

“…Let k ≥ 2 be fixed. As in [6] one can show the existence of (r, k)-colourings of K 2 r +1 with arbitrarily long odd girth. Let f ′ be the smallest integer such that there is an (f ′ , k)-colouring ϕ 1 of K 2 f ′ +1 with odd girth strictly greater than ℓ.…”

“…The next result provides, in particular, the lower bound for (11). We remark that for k fixed, and r large enough, the bounds from Theorem 6.2 can be improved, based on recent results from [6] (see Proposition 7.4). Theorem 6.2.…”

“…The Bondy-Erdős conjecture states that r r (C ℓ ) = 2 r−1 (ℓ − 1) + 1 for every odd ℓ ≥ 3. Recently, Jenssen and Skokan [16] proved that the Bondy-Erdős conjecture holds for fixed r and sufficiently large odd n. However, Day and Johnson [6] disprove the Bondy-Erdős conjecture by showing that for every odd ℓ ≥ 3 exist ε > 0 and sufficiently large r such that r r (C ℓ ) > (2 + ε) r−1 (ℓ − 1). We can imitate their construction to see that, analogously: Proposition 7.4.…”

“…We include a sketch of the proof of Proposition 7.4 for readers who are familiar with the construction of Day and Johnson in [6].…”

“…Let ϕ 2 be the (c + 1, k)-colouring of the complete graph on 2 ⌊c/k⌋ (ℓ − 1) vertices avoiding a monochromatic C ℓ , as given by Theorem 6.2. If r = mf ′ +c, with f ′ > c ≥ 0, then, following the construction in [6], one can define an (r, k)-colouring of the complete graph on (2 f ′ + 1) m • 2 ⌊c/k⌋ (ℓ − 1) > (2 + ε) r−f vertices.…”