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arXiv · 1703.02473

An improved lower bound for Folkman's theorem

József Balogh·Sean Eberhard·Bhargav Narayanan·Andrew Treglown·Adam Zsolt Wagner
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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 $k\in \mathbb{N}$.

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József Balogh, Sean Eberhard, Bhargav Narayanan, Andrew Treglown, Adam Zsolt Wagner. 2017-03-07. An improved lower bound for Folkman's theorem. https://doi.org/10.1112/blms.12058

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