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A canonical Ramsey theorem for exactly $m$-coloured complete subgraphs
- Publication Year :
- 2013
-
Abstract
- Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an $m$-coloured complete subgraph for every natural number $m$ or there exists an infinite subset $X \subset \mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on $X$ or there exists a distinguished vertex $v$ in $X$ such that $X \setminus \lbrace v \rbrace$ is $1$-coloured and each edge between $v$ and $X \setminus \lbrace v \rbrace$ has a distinct colour (all different to the colour used on $X \setminus \lbrace v \rbrace$). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding $m$-coloured complete subgraphs in colourings with finitely many colours.<br />Comment: 16 pages, improved presentation, fixed misprints, Combinatorics, Probability and Computing
- Subjects :
- Mathematics - Combinatorics
05D10 (Primary) 05C63 (Secondary)
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1303.2997
- Document Type :
- Working Paper