On the random greedy F-free hypergraph process

TL;DR

This paper analyzes a random greedy process for constructing large hypergraphs free of a fixed subhypergraph F, showing it produces near-optimal size hypergraphs with high probability.

Contribution

It provides asymptotic bounds on the size of hypergraphs generated by the random greedy F-free process, extending understanding of hypergraph extremal constructions.

Findings

Process terminates with hypergraphs of size O(n^{k-(|F|-k)/(e(F)-1)})

Bounds are tight up to logarithmic factors

Results apply to strictly k-balanced hypergraphs with certain properties

Abstract

Let $F$ be a strictly $k$-balanced $k$-uniform hypergraph with $e(F)\geq |F|-k+1$ and maximum co-degree at least two. The random greedy $F$-free process constructs a maximal $F$-free hypergraph as follows. Consider a random ordering of the hyperedges of the complete $k$-uniform hypergraph $K_n^k$ on $n$ vertices. Start with the empty hypergraph on $n$ vertices. Successively consider the hyperedges $e$ of $K_n^k$ in the given ordering, and add $e$ to the existing hypergraph provided that $e$ does not create a copy of $F$. We show that asymptotically almost surely this process terminates at a hypergraph with $\tilde{O}(n^{k-(|F|-k)/(e(F)-1)})$ hyperedges. This is best possible up to logarithmic factors.