... Rothschild ([GLR]).Theorem 1.3 Let r, n, t ∈ N. There exists N = N (r, n, t) such that for any r-coloringof the n-spaces of ANthere exists a t-space V such that the family of n-spaces containedin ... words, the function γ restricted to the standard n2-variable submatrices of v isthe lift of an r-coloring γof the setF({1, ,L})n<. By the choice of L, there thusexist non-empty sets ... standard n2-variable submatrices areγ-monochromatic.Theorem 1.7 extends the Bergelson-Leibman coloring theorem in the sense that ifone defines zero-variable matrices to be matrices with entries...