A note on balanced independent sets in the cube

Ramras conjectured that the maximum size of an independent set in the discrete cube Qn containing equal numbers of sets of even and odd size is 2n−1 − ( n−1 (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. The discrete hypercube Qn is the graph with vertices the subsets of [n] = {1, . . . , n} and edges between sets whose symmetric difference contains a single element. The cube Qn is bipartite, with classes X0 and X1 consisting of the sets of even and odd size respectively. The maximum-sized independent sets in Qn are precisely X0 and X1. Ramras [3] asked: how large an independent set can we find with half its elements in X0 and half in X1? Call such an independent set balanced. The following result verifies the conjecture made by Ramras for the case where n is odd. Theorem 1. The largest balanced independent set in Qn has size 2n−1 − 2 ( n − 2 (n − 2)/2 ) if n is even, 2n−1 − ( n − 1 (n − 1)/2 ) if n is odd. For a set A of vertices of Qn, write N(A) for the set of vertices adjacent to an element of A. The maximal independent sets in Qn all have the form A∪(X1\N(A)) for some A ⊆ X0. So for a maximum-sized balanced independent set we seek the largest A ⊆ X0 for which |A| ≤ |X1 \ N(A)|.