On Weak $\epsilon$-Nets and the Radon Number

Abstract

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak $\epsilon$-nets.

Appendices

Radon, Helly and VC

Here we prove that the Radon number bounds from above both the Helly number and the VC dimension (Lemma 1.11). The proof follows from the following two claims. Levi  [25] proved that

Claim A.1

Let C be a convexity space. If the Radon number of C is r then its Helly number is smaller than r.

Proof

(for completeness) Let $C' \subseteq C$ be a finite family such that $\bigcap _{c \in C'} c = \emptyset$. Let $K\subseteq C'$ be a minimal subfamily such that $\bigcap _{c\in K} c = \emptyset$. Assume towards a contradiction that $| K | \ge r$. Minimality implies that for each $k\in K$ we have $C_k : = \bigcap _{c\in K \setminus \{k\}} c \ne \emptyset$. Let $x_k \in C_k$. The $x_k$’s must be distinct (otherwise $\bigcap _{c\in K} c \ne \emptyset$). Thus, there is a partition of $\{x_k : k\in K\}$ into two parts $Y_1,Y_2$ such that ${\text {conv}}Y_1\cap {\text {conv}}Y_2\ne \emptyset$. But

$$\begin{aligned} {\text {conv}}Y_1\,\subseteq \!\! \bigcap _{k \in K : x_k\in Y_2} \!\!k \qquad \text {and} \qquad {\text {conv}}Y_2\,\subseteq \!\! \bigcap _{k \in K :x_k \in Y_1}\!\! k, \end{aligned}$$

by construction. This is a contradiction, so $|K|<r$. $\square$

We observe that

Claim A.2

Let C be a convexity space and B be its half-spaces. If the Radon number of C is r then the VC dimension of B is smaller than r.

Proof

Let $Y \subseteq X$ be of size r. The set Y can thus be partitioned into $Y_1,Y_2$ so that ${\text {conv}}Y_1\cap {\text {conv}}Y_2 \ne \emptyset$. Assume, towards a contradiction, that there is $b \in B$ such that $b \cap Y = Y_1$. Since B consists of half-spaces,⁹ there is $\bar{b} \in B \subseteq C$ such that $Y_2 = \bar{b} \cap Y$. This implies that ${\text {conv}}Y_1\cap {\text {conv}}Y_2= \emptyset$, which is a contradiction. Thus, for all $b \in B$ we have $b \cap Y \ne Y_1$ which means that the VC dimension is less than $|Y|=r$. $\square$

The Chromatic Number of the Kneser Graph

Here we prove a lower bound on the chromatic number of the Kneser graph (which is weaker than Lovász’s). We follow an argument of Alon (see footnote 7), who informed us that a similar argument was independently found by Szemerédi. We focus on a particular case, but the argument applies more generally.

Theorem B.1

For n being divisible by 4 we have $\chi (KG_{n,n/4}) > n/10$.

The first step in the proof is the following lemma proven by Kleitman  [24]. A family $F \subseteq 2^X$ is called intersecting if $f \cap f' \ne \emptyset$ for all $f,f' \in F$.

Lemma B.2

If $F_1,\ldots ,F_s \subset 2^{[n]}$, where each $F_i$ is intersecting, then

$$\begin{aligned} \biggl | \bigcup _{i \in [s]} F_i \biggr | \le 2^n-2^{n-s}. \end{aligned}$$

The lemma can be proven by induction on s. The case $s=1$ just says that an intersecting family has size at most $2^{n-1}$. The induction step is based on correlation of monotone events (for more details see, e.g.,  [6]).

Proof of Theorem B.1

Consider a proper coloring of $KG_{n,n/4}$ with s colors. Let $V_1,\ldots ,V_s$ be the partition of the vertices into color classes. Each $V_i$ is an intersecting family. Let $F_i$ be the family of sets $u \subseteq [n]$ that contain some set in $V_i$. Each $F_i$ is also intersecting. By the lemma above,

$$\begin{aligned} 2^n -\biggl | \bigcup _{i \in [s]} F_i \biggr | \ge 2^{n-s}. \end{aligned}$$

On the other hand, the complement of $\bigcup _{i \in [s]} F_i$ is of size less than

$$\begin{aligned} \sum _{k=0}^{n/4}{n \atopwithdelims ()k} \le 2^{n H(1/4)}, \end{aligned}$$

where $H(p) = - p \log p-(1-p) \log {(1-p)}$ is the binary entropy function. Hence, $s > n(1-H(1/4)) \ge n /10$. $\square$