Results in Borel chromatic numbers using Infinite games and Borel determinacy

Abstract: (Based on the paper “A determinacy approach to Borel combinatorics” by Andrew Marks) 
A Borel graph on a standard Borel space X is a symmetric irreflexive relation G on X that is Borel as a subset of X×X.

A Borel coloring of a Borel graph G on X is a Borel function c from X to a standard Borel space Y such that if
xGy then c(x)!=c(x)
The Borel chromatic number χB(G) of G is the least cardinality of a standard Borel space Y such that G has a Borel coloring with codomain Y.

To each marked group Γ and a standard Borel space X we can match a Borel graph G(Γ,X) 

We will use Boreal determinacy to prove that
χB(G(Γ*Δ),N)>=χB(G(Γ,N))+χB(G(Δ,N))-1