We study a natural generalization of the classical $\epsilon$-net problem (Haussler--Welzl 1987), which we call the "$\epsilon$-$t$-net problem": Given a hypergraph on $n$ vertices and parameters $t$ and $\epsilon\geq \frac t n$, find a minimum-sized family $S$ of $t$-element subsets of vertices such that each hyperedge of size at least $\epsilon n$ contains a set in $S$. When $t=1$, this corresponds to the $\epsilon$-net problem. We prove that any sufficiently large hypergraph with VC-dimension $d$ admits an $\epsilon$-$t$-net of size $O(\frac{ (1+\log t)d}{\epsilon} \log \frac{1}{\epsilon})$. For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of $O(\frac{1}{\epsilon})$-sized $\epsilon$-$t$-nets. We also present an explicit construction of $\epsilon$-$t$-nets (including $\epsilon$-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of $\epsilon$-nets (i.e., for $t=1$), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.

For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if every hyperedge coloring with $k$ colors of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous results by M. Axenovich, A. Gyárfás, H. Liu, and D. Mubayi [ Discrete Math., 322 (2014), pp. 69--77] and P. Erdös, A. Hajnal, A. Máté, and R. Rado, [ Combinatorial set theory: Partition Relations for Cardinals, Elsevier, Amsterdam, 1984] we show that $if r \geq 3 and n \nrightarrow (s)_k^r, then 2^n \nrightarrow (s+1)_{k+3}^{r+1}.$ This improves some of the known lower bounds on multicolor hypergraph Ramsey numbers. Given a hypergraph $H=(V,E)$, we consider the Ramsey-like problem of coloring all $r$-subsets of $V$ such that no hyperedge of size $\geq r+1$ is monochromatic. We provide upper and lower bounds on the number of colors necessary in terms of the chromatic number $\chi(H)$. In particular we show that this number is $O(\log^{(r-1)} (r \chi(H)) + r)$, where $\log^{y}$ is the $\log$ function applied $y$ times.

We study a natural generalization of the classical $ε$-net problem (Haussler--Welzl 1987), which we call the "$ε$-$t$-net problem": Given a hypergraph on $n$ vertices and parameters $t$ and $ε\geq \frac t n$, find a minimum-sized family $S$ of $t$-element subsets of vertices such that each hyperedge of size at least $εn$ contains a set in $S$. When $t=1$, this corresponds to the $ε$-net problem. We prove that any sufficiently large hypergraph with VC-dimension $d$ admits an $ε$-$t$-net of size $O(\frac{ (1+\log t)d}ε \log \frac{1}ε)$. For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of $O(\frac{1}ε)$-sized $ε$-$t$-nets. We also present an explicit construction of $ε$-$t$-nets (including $ε$-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of $ε$-nets (i.e., for $t=1$), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.

Given a set system $$(X, \mathcal {R})$$ such that every pair of sets in $$\mathcal {R}$$ have large symmetric difference, the Shallow Packing Lemma gives an upper bound on $$|\mathcal {R}|$$ as a function of the shallow-cell complexity of $$\mathcal {R}$$ . In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system $$(X, \mathcal {R})$$ with shallow-cell complexity $$\varphi (\cdot , \cdot )$$ and a parameter $$\epsilon > 0$$ , there exists a collection, called an $$\epsilon $$ -Mnet, consisting of $$O\bigl ( \frac{1}{\epsilon } \,\varphi \bigl ( O\bigl (\frac{1}{\epsilon } \bigr ), O(1)\bigr ) \bigr )$$ subsets of X, each of size $$\Omega ( \epsilon |X| )$$ , such that any $$R \in \mathcal {R}$$ with $$|R| \ge \epsilon |X|$$ contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal $$\epsilon $$ -net bound follows.

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