A note on balanced independent sets in the cube

TL;DR

This paper proves Ramras's conjecture on the maximum size of balanced independent sets in the discrete cube, providing exact bounds for both odd and even dimensions using isoperimetric inequalities.

Contribution

It confirms Ramras's conjecture and derives bounds for even dimensions, advancing understanding of independent sets in the discrete cube.

Findings

Confirmed Ramras's conjecture for odd n

Derived bounds for even n

Connected independent set size to isoperimetric inequalities

Abstract

Ramras conjectured that the maximum size of an independent set in the discrete cube containing equal numbers of sets of even and odd size is 2^(n-1) - (n-1 choose (n-1)/2) when n is odd. We prove this conjecture, and find the analogous bound when n is even. The result follows from an isoperimetric inequality in the cube.