# Probabilistic Existence of Large Sets of Designs

## TL;DR

This paper extends a probabilistic technique to prove the existence of large sets of combinatorial designs, specifically showing they exist for large parameters when certain divisibility conditions are met, resolving a longstanding conjecture.

## Contribution

It adapts a recent probabilistic method to establish the existence of large sets of $t$-designs, proving their existence for all sufficiently large $n$ when $k > 12t$ and divisibility conditions hold.

## Key findings

- Large sets of $t$-$(n,k,)$ designs exist for sufficiently large $n$ when $k > 12t$.
- The method confirms the existence of these designs with positive probability.
- It resolves the conjecture for large sets of designs under specified conditions.

## Abstract

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been recentlyintroduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t$-$(n,k,\lambda)$ combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a $t$-$(n,k,\lambda)$ design exist but, in fact, with positive probability there exists a large set of such designs -- that is, a partition of the set of $k$-subsets of $[n]$ into $t$-designs $t$-$(n,k,\lambda)$ designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large $n$, large sets of $t$-$(n,k,\lambda)$ designs exist whenever $k > 12t$ and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all $k > 12t$.