Nash-Williams barrier partition theorem

Every finite coloring of a barrier in better-quasi-order theory has an infinite restriction on which all barrier members have the same color. It extends finite-dimensional Ramsey's theorem from fixed-length subsets to variable-length initial segments. A proof proceeds by well-founded induction on the initial-segment tree and a fusion construction selecting successively nested infinite tails. It implies that every finite set with the equality order is a better-quasi-ordering. The barrier/front partition proof is given in Section 3.2 of arxiv.org/pdf/1604.05866 .