A Sharper Ramsey Theorem for Constrained Drawings

Given a graph $G$ and a collection $C$ of subsets of $R^d$ 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 $C$, in such a way that nonadjacent 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 results can be used to obtain Helly-type theorems. More precisely, we prove the following. For each $n$ and $b$, there is $N=O\left(b^{2n-3}\right)$ with the following properties: If $G$ is a drawing of a graph on $N$ vertices and $C$ is a collection of sets of $R^d$ such that each $\left(b+1\right)$-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 $h\left(b\right)$ of degree at most $2d+3$ such that the following holds. For every family $ℱ$ of sets in $R^d$, its Helly number is at most $h\left(b\right)$, provided that the intersection of any nonempty subfamily has at most $b$ path-connected components, and trivial homology groups $H_1$, $H_2,\dots . H_{\lceil d∕2\rceil -1}$. This dramatically improves the original theorem by Matoušek which had stronger assumption and a tower-like bound on $h\left(b\right)$. Under the same assumptions, our technique can also be used to bound Radon numbers.