Probabilistic existence of regular combinatorial structures

TL;DR

This paper proves the probabilistic existence of various regular combinatorial structures like orthogonal arrays and t-designs, providing size optimality and counting estimates using advanced probabilistic methods.

Contribution

It introduces a probabilistic method to establish the existence of regular combinatorial objects with optimal sizes and counts, previously unknown.

Findings

Existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations.

Provides size estimates that are optimal up to polynomial factors.

Develops a local central limit theorem for lattice random walks to support the proofs.

Abstract

We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen structure has the required properties with positive yet tiny probability. Our method allows also to give rather precise estimates on the number of objects of a given size and this is applied to count the number of orthogonal arrays, t-designs and regular hypergraphs. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.