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Finding Partite Hypergraphs Efficiently

Ferran Espuña

TL;DR

The value obtained 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.

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 Turán on bipartite graphs. The value we obtain for the part size matches the order of magnitude guaranteed by the non-constructive proof due to Erdős and is tight up to a constant factor.

Finding Partite Hypergraphs Efficiently

TL;DR

The value obtained 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.

Abstract

We provide a deterministic polynomial-time algorithm that, for a given -uniform hypergraph with vertices and edge density , finds a complete -partite subgraph of with parts of size at least . This generalizes work by Mubayi and Turán on bipartite graphs. The value we obtain for the part size matches the order of magnitude guaranteed by the non-constructive proof due to Erdős and is tight up to a constant factor.

Paper Structure

This paper contains 5 sections, 3 theorems, 20 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. The algorithm
  3. Proof of correctness
  4. Proof of Polynomial Complexity
  5. Concluding Remarks and Future Work

Key Result

Theorem 1.2

There is a deterministic algorithm that, given a $k$-uniform hypergraph $H$ with $n$ vertices and $m=d \binom{n}{k}$ edges, finds a complete balanced $k$-partite subgraph $K(t, \overset{k}{\dots}, t)$ in polynomial time, where

Theorems & Definitions (5)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Theorem 3.2
  • proof