# An improved lower bound for Folkman's theorem

## TL;DR

This paper improves the lower bound for Folkman's theorem, demonstrating that the minimal number ensuring monochromatic sumsets in two-colourings grows doubly exponentially with respect to the size of the set.

## Contribution

The authors establish a significantly stronger lower bound for Folkman's number, advancing the understanding of combinatorial colorings and sumset properties.

## Key findings

- New lower bound: F(k) ≥ 2^{2^{k-1}/k}
- Doubly exponential growth of Folkman's number
- Improvement over previous bound by Erdős and Spencer

## 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 $k\in \mathbb{N}$.