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An improved lower bound for Folkman's theorem
- Publication Year :
- 2017
-
Abstract
- Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the sums of the form $\sum_{x \in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same colour. In 1989, Erd\H{o}s and Spencer showed that $F(k) \ge 2^{ck^2/ \log k}$, where $c >0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \ge 2^{2^{k-1}/k}$ for all $k\in \mathbb{N}$.<br />Comment: 5 pages, Bulletin of the LMS
- Subjects :
- Mathematics - Combinatorics
05D10 (Primary), 05D40 (Secondary)
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1703.02473
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1112/blms.12058