Colorful Helly via induced matchings

In 1913, Helly famously proved that for every collection of convex sets $C_{1},\ldots,C_{n}$ in $\mathbb{R}^{d}$ with an empty intersect, there exists a set $S\subset[n]$ of size $|S|\leqslant d+1$ with $\bigcap_{i\in S}C_{i}=\emptyset$. Moreover, $d+1$ is the smallest number with this property. This result is known as Helly’s theorem and is fundamental result in combinatorial geometry, with a large number of generalizations and applications over the years. See for example [5], [9], [12], [24], and the references therein. We also recommend the surveys [3] and [6], focusing on more recent developments. To state some of these different variants of Helly’s theorem, it is often standard to introduce the following terminology. For a set $X$ and a family $\mathscr{F}$ of subsets of $X$, the Helly number $h=h(X,\mathscr{F})\in\mathbb{N}$ of $\mathscr{F}$ is defined as the smallest integer $h$ such that for any finite subfamily $K$ of $\mathscr{F}$, if every subset of at most $h$ members of $\mathscr{F}$ has a nonempty intersection then all sets in $K$ have a nonempty intersection. If no such $h$ exists, then $h(X,\mathscr{F}):=\infty$. Helly’s theorem above then asserts that the Helly number of the family of convex sets in $\mathbb{R}^{d}$ is $d+1$. Another example of a known Helly number is the Helly number of spheres in $\mathbb{R}^{d}$. In [23], Maehara proved that this equals $d+2$. An additional example is the theorem of Doignon [11] which asserts that the Helly number of convex lattice sets in $\mathbb{R}^{d}$, that is sets of the form $C\cap\mathbb{Z}^{d}$ where $C$ is a convex set in $\mathbb{R}^{d}$ is $2^{d}$. A combinatorial example is the fact that the Helly number of the collection of (sets of vertices of) subtrees of any given tree is $2$.

An algebraic example that in some sense motivated the present paper is the fact that for any field $\mathbb{F}$ the Helly number of hypersurfaces of degree at most $D$ in $\mathbb{F}^{d}$ is ${D+d\choose d}$. This was proved in 1986 by Deza and Frankl [10] in the following elegant way: for any set of polynomials $f_{1},\ldots,f_{n}$ of degree $\leqslant D$ in $\mathbb{F}[x_{1},\ldots,x_{d}]$ with the property that their zero sets $Z(f_{i}):=\left\{x\in\mathbb{F}^{d}:f_{i}(x)=0\right\}$ satisfy $\bigcap_{i=1}^{n}Z(f_{i})=\emptyset$, consider a minimal subset $S\subset[n]$ such that $\bigcap_{i\in S}Z(f_{i})=\emptyset$. By definition, this is a set $S$ which must also satisfy the property that for every $j\in S$, the intersection $\bigcap_{i\in S\setminus\{j\}}Z(f_{i})$ is nonempty. In other words, for every $j\in S$ there exists $x_{j}\in\mathbb{F}^{d}$ such that:

By a (nowadays) standard argument, it is not difficult to see that this condition implies that the set of polynomials $\left\{f_{i}\right\}_{i\in S}$ is a set of linearly independent polynomials in $\mathbb{F}[x_{1},\ldots,x_{d}]$. This yields the estimate $|S|\leqslant\binom{D+d}{d}$, since $\binom{D+d}{d}$ equals precisely the dimension of the vector space of polynomials of degree at most $D$ in $\mathbb{F}[x_{1},\ldots,x_{d}]$. In [13], Frankl subsequently extended this argument to varieties of bounded degree. It is perhaps instructive to highlight at this point that a similar strategy cannot be used to prove Helly’s original theorem for convex sets. In particular, if $d\geqslant 2$ and $C_{1},\ldots,C_{n}$ is a collection of convex sets in $\mathbb{R}^{d}$ such that $\bigcap_{i=1}^{n}C_{i}=\emptyset$, then it is no longer true that the size of a minimal set $S\subset[n]$ with the property that $\bigcap_{i\in S}C_{i}=\emptyset$ has to be bounded independently of $n$ (rather than a minimum set). Indeed, note that while for every $j\in S$ there exists a point $x_{j}\in\mathbb{R}^{d}$ such that

this condition no longer implies that $|S|=O_{d}(1)$ (let alone $|S|\leqslant d+1$).¹¹1Indeed, note that for any $d\geqslant 2$ and any arbitrarily large $m\geqslant 1$ one can construct $m$ convex sets $C_{1},\,\ldots,\,C_{m}\in\mathbb{R}^{d}$ such that for every $i\in[m]$ there exists $x_{i}$ such that $x_{i}\in\bigcap_{j\neq i}C_{j}\setminus C_{i}$. Let $P$ be a convex polytope with at least $m$ facets. Pick $m$ facets $f_{1},\,\ldots,\,f_{m}$ and let $C_{i}$ be the half-space bounded by the hyperplane supporting $f_{i}$ and containing $P$. Let $x_{i}$ be any point in the polytope determined by the hyperplane supporting $f_{i}$, and the hyperplane supporting all the neighbor facets of $f_{i}$. By construction, $x_{i}\in C_{j}$ if and only if $i\neq j$. Nevertheless, this type of argument turns out to be quite powerful in settings where conditions like (1) or (2) do generate a set of $|S|$ linear independent objects in a vector space of bounded dimension. For example, in a recent nice paper [1], Alon, Jin, and Sudakov also used this strategy to determine that the Helly number of the set of Hamming balls of radius $t$ inside an $n$-dimensional space $X^{n}$ is $2^{t+1}$ (where $X$ is an arbitrary finite or infinite set).

In this paper, we generalize this story by showing that in a set system exhibiting conditions like (1) or (2) one can also establish an optimal colorful Helly theorem. To state things more precisely going forward, it will be convenient to consider the following abstract setting. Let $X$ be a set and let $\mathscr{F}$ be a collection of subsets of elements from $X$. We define the colorful Helly number of the set system $(X,\mathscr{F})$ to be the minimum positive integer $\eta=\eta(X,\,\mathscr{F})$ for which the following statement holds: if finite subfamilies $\mathscr{F}_{1},\,\ldots,\,\mathscr{F}_{\eta}\subset\mathscr{F}$ are such that $\cap_{F\in\mathscr{F}_{i}}F=\emptyset$ for every $i=1,\,\ldots,\,\eta$, then there exists $F_{i}\in\mathscr{F}_{i}$ such that $F_{1}\cap\cdots\cap F_{\eta}=\emptyset$.

Like before, if no such $\eta$ exists, then we say $\eta(X,\,\mathscr{F}):=\infty$. It is a celebrated theorem of Lovász (whose proof first appeared in a paper by Bárány [4]) that if $X=\mathbb{R}^{d}$ and $\mathscr{F}$ is the family of convex sets in $\mathbb{R}^{d}$ then $\eta(X,\,\mathscr{F})=d+1$. This generalization of Helly’s theorem has managed to inspire a lot of further interesting research directions on its own, due to its rather mysterious geometric nature. For example, while clearly $h(X,\mathscr{F})\leqslant\eta(X,\mathscr{F})$, it is not always the case that $h(X,\mathscr{F})=\eta(X,\mathscr{F})$. Let $X=\mathbb{R}^{d}$ and let $\mathscr{F}$ consist of a family of axis-parallel boxes: it is a folklore result that in this case $h(X,\mathscr{F})=2$, while it is not difficult to see that $\eta(X,\mathscr{F})=d+1$.²²2Lovász’s theorem implies $\eta(X,\mathscr{F})\leqslant d+1$, whereas $\eta(X,\mathscr{F})\geqslant d+1$ follows by considering the $d$-coloring of the facets of a hypercube in $\mathbb{R}^{d}$ so that each opposite pair of facets carry the same color. This is a coloring with $d$ color classes $\mathscr{F}_{1},\ldots,\mathscr{F}_{d}$ where $\cap_{F\in\mathscr{F}_{i}}F=0$ for every $i=1,\,\ldots,\,\eta$, but such that $F_{1}\cap\cdots\cap F_{d}\neq\emptyset$ for every $F_{i}\in\mathscr{F}_{i}$. It is therefore natural to ask which Helly-type theorems admit appropriate colorful versions (see for example the survey of Amenta, De Loera, and Soberón [3, Section 2.2] for more context).

Our results below address this question by providing several natural sufficient conditions for the existence of a colorful Helly theorem. We will work with the incidence graph of $X$ and $\mathscr{F}$, which is the bipartite graph $G_{X,\,\mathscr{F}}$ on $X\cup\mathscr{F}$ where $(x,\,F)\in X\times\mathscr{F}$ is an edge if $x\in F$. The bipartite complement $\overline{G}_{X,\,\mathscr{F}}$ of $G_{X,\,\mathscr{F}}$ denotes the graph with vertex set $X\cup\mathscr{F}$ where $(x,\,F)\in X\times\mathscr{F}$ is an edge if $x\not\in F$. We will also be concerned with the size $\tau=\tau(X,\,\mathscr{F})$ of the maximum induced matching inside $\overline{G}_{X,\,\mathscr{F}}$. We shall sometimes refer to a matching in $\overline{G}_{X,\,\mathscr{F}}$ as a comatching of the set system $(X,\mathscr{F})$, and to $\tau(X,\,\mathscr{F})$ as the size of the largest comatching in $(X,\mathscr{F})$ (or simply, the comatching number of $(X,\mathscr{F})$). We are now ready to state our first result.

As we will see in Section 2, this result is optimal and even the additive $+1$ term is necessary. Furthermore, in certain situations we can remove the assumption that the $\mathscr{F}_{i}$’s are finite. Let $\eta^{\infty}(X,\mathscr{F})$ be the colorful Helly number with the finiteness assumption on $\mathscr{F}_{i}$ removed. Clearly, we have $\eta(X,\mathscr{F})\leqslant\eta^{\infty}(X,\mathscr{F})$.

We say $\mathscr{F}$ is Noetherian if for any sequence of sets $F_{1},F_{2},\ldots$ in $\mathscr{F}$, the descending chain of sets

stabilizes. For example, we have

• •

  If $X$ is finite, then $\mathscr{F}$ is Noetherian.

• •

  If $X$ is $\mathbb{C}^{n}$ or $\mathbb{R}^{n}$, and $\mathscr{F}$ is any family of algebraic varieties in $X$, then $\mathscr{F}$ is Noetherian due to the Nullstellensatz and the fact that $\mathbb{C}[x_{1},\ldots,x_{n}]$ is a Noetherian ring.

• •

  If $X$ is $\mathbb{R}^{2}$, and $\mathscr{F}$ is the family of convex sets, then $\mathscr{F}$ not Noetherian. We can check that $\eta(X,\mathscr{F})=3$ but $\eta^{\infty}(X,\mathscr{F})=\infty$.

If $\mathscr{F}$ is Noetherian, then the finiteness assumption on $\mathscr{F}_{i}$ is not necessary. While this observation is not hard to prove, we are unaware of any literature that discusses this subtlety.

In addition, we can consider a slightly stronger notion of $\tau(X,\mathscr{F})$. Let $\tau^{\prime}(X,\mathscr{F})$ be the largest $\tau^{\prime}$ such that there exist $x_{1},\ldots,x_{\tau^{\prime}+1}\in X$ and $F_{1},\ldots,F_{\tau^{\prime}}$ such that $x_{i}\in F_{j}$ if and only if $i\neq j$. We call this a comatching with intersection, and $x_{\tau^{\prime}+1}$ the common point. Clearly, we have

Using this terminology, we will show the following refinement of Theorem 1.1, which is in some sense our main result.

Theorem 1.3 has a few new consequences, which we will discuss in detail in Section 3. First, we will argue that Theorem 1.3 implies that the colorful Helly number for $d$-dimensional spheres in $\mathbb{R}^{d}$ is $d+2$, generalizing the work of Maehara [23]. Second, we will show that Theorem 1.3 establishes an optimal colorful Helly theorem for any set of hypersurfaces of bounded degree, generalizing the works of Motzkin [26] and Deza-Frankl [10]. Last but not least, we will show that the colorful Helly number of the set of Hamming balls of radius $t$ inside an $n$-dimensional space $X^{n}$ is $2^{t+1}$, generalizing the work of Alon, Jin, and Sudakov [1]. Turns out, it is also possible to prove the first two consequences of Theorem 1.3 using some more ad-hoc arguments. These arguments have some shortcomings, but we considered them to be quite instructive so we will discuss them in Section 2.

It is also natural to wonder whether set systems with bounded comatching number satisfy a so-called fractional Helly theorem. In the context of Helly’s theorem for convex sets, this theorem was first established by Katchalski and Liu in [19]: if $\mathscr{F}$ is a family of $n\geqslant d+1$ convex sets in $\mathbb{R}^{d}$ such that the number of $(d+1)$-tuples of $\mathscr{F}$ with non-empty intersection is at least $\alpha{n\choose d+1}$ for some constant $\alpha>0$, then there are at least $\beta n$ members of $\mathscr{F}$ whose intersection is non-empty, where $\beta>0$ is a constant which depends only on $\alpha$ and $d$. The optimal choice $\beta=1-(1-\alpha)^{\frac{1}{d+1}}$ was determined by Kalai [17]. In [15], Holmsen showed that the the fractional Helly theorem can be derived as a purely combinatorial consequence of the colorful Helly theorem, and then later Holmsen and Lee [16] used this result to prove that a fractional Helly theorem holds in all set systems with a bounded Radon number. We note that an abstract fractional Helly theorem for sets with bounded comatching number also holds.

While it is also possible to use the result [15] and derive a fractional Helly theorem using Theorem 1.3, we note that the statement from Corollary 1.4 (with the correct dependence on $\tau$) follows from the work of Matoušek [25]. Matoušek showed that the fractional Helly number of a set system is at most the dual VC dimension plus one. On the other hand, the dual VC-dimension of a set system is dominated by our parameter $\tau^{\prime}$: Suppose we are given a family $A\subset\mathscr{F}$ which is dual shattered by $X$, meaning, for every $B\subset A$, there exists $x\in\bigcap_{b\in B}B\setminus\bigcap_{a\in A}A$. Taking $B:=A\setminus\{a\}$ for $a\in A$ and $B:=A$ yields that $A$ is a comatching with intersection.

We thank Qiuyu Ren for numerous helpful discussions, and proving the $d=2$ case of Proposition 4.2.

In this section, we discuss some preliminary results that only require ad-hoc arguments (in hindsight). The first one concerns a colorful version of Helly’s theorem for spheres.

We begin with the proof of Theorem 1.3, which subsumes Theorem 1.1.