arXiv · math/0407235
This note gives a short proof on characterizations of a forest to be equitably k-colorable.
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Gerard J. Chang. 2004-07-14. A note on equitable colorings of forests. https://arxiv.org/abs/math/0407235
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Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In this paper, we confirm the conjecture when $r$ is a power of prime. In particular, the bound we obtain is tight.
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