Abstract
The Ramsey number r(Cℓ,Kn) is the smallest natural number N such that every red/blue edge colouring of a clique of order N contains a red cycle of length ℓ or a blue clique of order n. In 1978, Erdős, Faudree, Rousseau, and Schelp conjectured that r(Cℓ,Kn) = (ℓ−1)(n−1)+1 for ℓ ≥ n ≥ 3 provided (ℓ, n) ≠ (3,3).
We prove that, for some absolute constant C ≥ 1, we have r(Cℓ,Kn) = (ℓ−1)(n−1)+1 provided ℓ ≥ C (log n / log log n). Up to the value of C this is tight since we also show that, for any ε > 0 and n > n0(ε), we have r(Cℓ,Kn) ≫ (ℓ−1)(n−1)+1 for all 3 ≤ ℓ ≤ (1−ε)(log n / log log n).
This proves the conjecture of Erdős, Faudree, Rousseau, and Schelp for large ℓ, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erdős, Faudree, Rousseau, and Schelp.
We prove that, for some absolute constant C ≥ 1, we have r(Cℓ,Kn) = (ℓ−1)(n−1)+1 provided ℓ ≥ C (log n / log log n). Up to the value of C this is tight since we also show that, for any ε > 0 and n > n0(ε), we have r(Cℓ,Kn) ≫ (ℓ−1)(n−1)+1 for all 3 ≤ ℓ ≤ (1−ε)(log n / log log n).
This proves the conjecture of Erdős, Faudree, Rousseau, and Schelp for large ℓ, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erdős, Faudree, Rousseau, and Schelp.
| Original language | English |
|---|---|
| Article number | rnz119 |
| Pages (from-to) | 275-300 |
| Number of pages | 26 |
| Journal | International Mathematics Research Notices |
| Volume | 2021 |
| Issue number | 1 |
| Early online date | 10 Jul 2019 |
| DOIs | |
| Publication status | E-pub ahead of print - 10 Jul 2019 |
Keywords
- Ramsey theory