= Nash-Williams barrier partition theorem
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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 https://arxiv.org/pdf/1604.05866 .
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