Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure

Bounding Helly Numbers via Betti Numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest

Bounding Radon Number via Betti Numbers

We prove general topological Radon-type theorems for sets in ?^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ?-nets as well as a (p,q)-theorem. More precisely: Let X be either ?^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ?? coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ?^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface

Bounding Radon's number via Betti numbers

We prove general topological Radon type theorems for sets in $\mathbb R^d$, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly and colorful Helly theorems, and consequently an existence of weak $\varepsilon$-nets as well as a $(p,q)$-theorem. More precisely: Let $X$ be either $\mathbb R^d$, smooth real $d$-manifold, or a finite $d$-dimensional simplicial complex. Then if $\mathcal F$ is a finite family of sets in $X$ such that $\widetilde\beta_i(\bigcap \mathcal G; \mathbb Z_2)$ is at most $b$ for all $i=0,1,\ldots, k$ and $\mathcal G\subseteq \mathcal F$, then the Radon's number of $\mathcal F$ is bounded in terms of $b$ and $X$. Here $k=\left\lceil\frac{d}{2}\right\rceil-1$ if $X=\mathbb R^d$; $k=d-1$ if $X$ is a smooth real $d$-manifold and not a surface, $k=0$ if $X$ is a surface and $k=d$ if $X$ is a $d$-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let $\mathcal F$ be a finite family of open sets in a surface $S$ such that for every $\mathcal G\subseteq \mathcal F$, $\bigcap \mathcal G$ is either empty, or path-connected. Then the fractional Helly number of $\mathcal F$ is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.Comment: 11 pages, 2 figure