A sharper Ramsey theorem for constrained drawings

Given a graph $G$ and a collection $\mathcal C$ of subsets of $\mathbb{R}^d$ indexed by the subsets of vertices of $G$, a constrained drawing of $G$ is a drawing, where each edge is drawn inside some set from $\mathcal C$, in such a way that non-adjacent edges are drawn in sets with disjoint indices. In this paper we prove a Ramsey type result for such drawings. Furthermore we show how the result can be used to obtain Helly type theorems. More precisely, we prove the following. For each $n$ and $b$, there is $N=O(b^{2n-3})$ with the following properties: If $G$ is a drawing of a graph on $N$ vertices and $\mathcal C$ is a collection of sets of $\mathbb{R}^d$ such that each $(b+1)$-tuple $T$ of vertices lies in a set indexed by $T$ and contains at least one edge in $T$, then in $G$, we can find a constrained copy of the complete graph $K_n$. As a direct consequence we obtain the following Helly type result: For each $d$, there is a polynomial $h(b)$ of degree at most $2d+3$ such that the following holds. For every family $\mathcal F$ of sets in $\mathbb{R}^d$, its Helly number is at most $h(b)$, provided that the intersection of any non-empty subfamily has at most $b$ path-connected components, and trivial homology groups $H_1$, $H_2$, .... $H_{\lceil d/2\rceil-1}$. This dramatically improves the original theorem by Matou\v{s}ek which had stronger assumption and a tower-like bound on $h(b)$. Under the same assumptions, our technique can also be used to bound Radon numbers.