Component sizes in the supercritical percolation on the binary cube

TL;DR

This paper provides a concise, self-contained proof of the well-known results concerning the sizes of components in supercritical percolation on high-dimensional binary cubes, revisiting classical findings with clarity.

Contribution

It offers a simplified, self-contained proof of established theorems on component sizes in supercritical percolation on the binary cube.

Findings

Confirmed classical results on component sizes in supercritical percolation

Provided a shorter, more accessible proof of existing theorems

Clarified the understanding of percolation behavior in high dimensions

Abstract

We present a relatively short and self-contained proof of the classical result on component sizes in the supercritical percolation on the high dimensional binary cube, due to Ajtai, Koml\'os and Szemer\'edi (1982) and to Bollob\'as, Kohayakawa and \L uczak (1992).