Remarks on the Box Problem

Abstract

Let Iijk be an N ×N ×N tensor of 1’s and 0’s. We say I satisfies the (E) condition provided it does not contain 1’s at all the corners of any nontrivial box. That is, I satisfies the (E) condition, if for any i1 = i2, j1 = j2, k1 = k2, we have that Π2l =1Π2 m=1Π2 n=1Iiljmkn = 0. We prove the following Theorem. Let p be any prime. There is a p3 − 1 × p3 − 1 × p3 − 1 tensor I satisfying the (E) condition containing p2(p3 − 1)2 many 1’s.