An extremal theorem in the hypercube

TL;DR

This paper establishes a broad class of subgraphs H in the hypercube for which the maximum edges in H-free subgraphs grow slower than the total edges, providing a unified method to prove such extremal results.

Contribution

It introduces a general approach to determine when the extremal number ex(Q_n, H) is o(e(Q_n)) for various subgraphs H in the hypercube.

Findings

ex(Q_n, H) = o(e(Q_n)) for a wide class of H

Unified proof technique for hypercube extremal problems

Specific result for cycles C_{2t} with t >= 4, t ≠ 5

Abstract

The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Q_n, H) = o(e(Q_n)). In particular, our method gives a unified approach to proving that ex(Q_n, C_{2t}) = o(e(Q_n)) for all t >= 4 other than 5.