Forbidden Configurations: Finding the number predicted by the Anstee-Sali Conjecture is NP-hard

TL;DR

This paper proves that determining the asymptotic number of edges in hypergraphs avoiding a fixed subhypergraph, as predicted by the Anstee-Sali conjecture, is an NP-hard problem, highlighting computational complexity challenges.

Contribution

It establishes the NP-hardness of computing the asymptotic behaviour of forb(m,F), assuming the Anstee-Sali conjecture, which was previously unproven.

Findings

Proves NP-hardness of finding asymptotics of forb(m,F)

Links conjecture validity to computational complexity

Highlights difficulty in hypergraph extremal problems

Abstract

Let F be a hypergraph and let forb(m,F) denote the maximum number of edges a hypergraph with m vertices can have if it doesn't contain F as a subhypergraph. A conjecture of Anstee and Sali predicts the asymptotic behaviour of forb(m,F) for fixed F. In this paper we prove that even finding this predicted asymptotic behaviour is an NP-hard problem, meaning that if the Anstee-Sali conjecture were true, finding the asymptotics of forb(m,F) would be NP-hard.