Uniform hypergraphs containing no grids

TL;DR

This paper constructs large linear hypergraphs that avoid grid and triangle configurations, providing near-optimal solutions and new bounds relevant to coding theory and combinatorial designs.

Contribution

It introduces constructions of large linear hypergraphs free of grids and triangles, advancing combinatorial design theory and coding applications.

Findings

Constructed large linear hypergraphs without grids for r ≥ 4

Developed hypergraphs avoiding both grids and triangles

Derived new bounds for superimposed codes and designs

Abstract

A hypergraph is called an r by r grid if it is isomorphic to a pattern of r horizontal and r vertical lines. Three sets form a triangle if they pairwise intersect in three distinct singletons. A hypergraph is linear if every pair of edges meet in at most one vertex. In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For r at least 4 our constructions are almost optimal. These investigations are also motivated by coding theory: we get new bounds for optimal superimposed codes and designs.