Core Concepts
By leveraging constructions from coding theory, we obtain improved lower bounds on the maximum number of edges in r-uniform hypergraphs with girth 5 and 6, for all r ≥ 3.
Abstract
The paper studies the maximum number of edges in r-uniform hypergraphs with girth 5 and 6, denoted by exr(N, C<5) and exr(N, C<6) respectively, where N is the number of vertices and C<g denotes the family of Berge cycles of length at most g-1.
Key highlights:
The authors address an unproved claim from prior work that the lower bound exr(N, C<5) = Ωr(N^{3/2-o(1)}) holds for all r ≥ 3. They identify an obstacle in the claimed proof and show that this obstacle can be overcome when r ∈ {4, 5, 6}.
For all other r, the authors use constructions from coding theory to prove new lower bounds on exr(N, C<5) and exr(N, C<6) that improve upon the previous probabilistic bounds.
The authors also show that recent results on hypergraph Turán problems can be used to improve the sphere packing bound for linear codes of distance 6.
The paper provides a comprehensive analysis of the connections between hypergraph Turán problems and coding theory, leading to new insights and improved bounds in both areas.