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A canonical Ramsey theorem for even cycles in random graphs
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 06 August 2026, pp. 1-27
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The celebrated canonical Ramsey theorem of Erdős and Rado implies that for
$2\leq k\in {\mathbb{N}}$, any colouring of the edges of 
$K_n$ with 
$n$ sufficiently large gives a copy of 
$C_{2k}$ which has one of three canonical colour patterns: monochromatic, rainbow or lexicographic. In this paper we show that if 
$p=\omega (n^{-1+1/(2k-1)}\log n)$, then 
${\mathbf G}(n,p)$ will asymptotically almost surely also have the property that any colouring of its edges induces canonical copies of 
$C_{2k}$. This determines the threshold for the canonical Ramsey property with respect to even cycles, up to a 
$\log$ factor.
