A note on balanced independent sets in the cube
  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. 
Ben Barber
math.CO