Finding Partite Hypergraphs Efficiently

TL;DR

This paper introduces a deterministic polynomial-time algorithm to find large complete k-partite subgraphs in k-uniform hypergraphs, generalizing bipartite graph results and matching theoretical bounds.

Contribution

It provides the first efficient algorithm for finding large partite subgraphs in hypergraphs, extending classical bipartite graph results to hypergraphs.

Findings

Algorithm runs in polynomial time

Finds complete k-partite subgraphs with optimal size bounds

Matches non-constructive theoretical guarantees

Abstract

We provide a deterministic polynomial-time algorithm that, for a given $k$-uniform hypergraph $H$ with $n$ vertices and edge density $d$, finds a complete $k$-partite subgraph of $H$ with parts of size at least ${c(d, k)(\log n)^{1/(k-1)}}$. This generalizes work by Mubayi and Tur\'{a}n on bipartite graphs. The value we obtain for the part size matches the order of magnitude guaranteed by the non-constructive proof due to Erd\H{o}s and is tight up to a constant factor.