# Pattern Avoidance Over a Hypergraph

## TL;DR

This paper investigates permutation pattern avoidance constrained by hypergraph structures, providing sharp bounds for random hypergraphs and those with large cliques, and introduces a supersaturation version of Furedi-Hajnal.

## Contribution

It introduces new bounds for permutation avoidance over hypergraphs and proves a supersaturation version of Furedi-Hajnal, advancing combinatorial understanding.

## Key findings

- Sharp bounds for permutation avoidance in random hypergraphs
- Bounds established for hypergraphs with large cliques
- A supersaturation version of Furedi-Hajnal proved

## Abstract

We consider the problem of bounding the number of permutations $\sigma\in S_n$ that avoid a fixed permutation $\pi\in S_k$ in specific indices given by a $k$-uniform hypergraph $\Lambda$. We obtain relatively sharp bounds in the case where $\Lambda$ is a random hypergraph, and find bounds in the case where $\Lambda$ contains many large cliques. Along the way, we prove a supersaturation version of F\"uredi-Hajnal, which may be of independent interest.