Let l,d,k be natural numbers. We want to prove that for large numbers n, for each k-coloring of the n-dimensional cube with side length l, there is a d-dimensional subspace that all of its vertices have the same color. Let H(l,d,k) be the least number such that for n≥H(l,d,k) the previus statement holds.
a) Prove that:
H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)}
b) Prove that
H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1)
c) Prove the statement of problem.
d) Prove Van der Waerden's Theorem. geometry3D geometrycombinatorics proposedcombinatorics