The existence of designs

TL;DR

This paper proves the existence of combinatorial designs under certain conditions, resolving a long-standing conjecture and extending results to clique decompositions in pseudorandom simplicial complexes.

Contribution

It establishes the existence conjecture for combinatorial designs and generalizes to clique decompositions with pseudorandomness and extendability conditions.

Findings

Proved the existence conjecture for combinatorial designs.

Extended results to clique decompositions in pseudorandom complexes.

Identified divisibility and extendability as key conditions for existence.

Abstract

We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. As a further generalisation, we obtain the same conclusion only assuming an extendability property and the existence of a robust fractional clique decomposition.