The existence of designs II

TL;DR

This paper generalizes the existence of combinatorial designs to subset sums in lattice structures associated with simplicial complexes, unifying various hypergraph decomposition problems and providing new approximate counting results.

Contribution

It introduces a broad framework for combinatorial design existence, encompassing hypergraph decompositions, and offers novel approximate counting techniques for complex structures.

Findings

Unified framework for design existence in lattices and simplicial complexes

Includes hypergraph decompositions with additional data like colors and orders

Provides new approximate counting results for high-dimensional permutations and Sudoku squares

Abstract

We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares.