I've been trying to calculate some known bounds on Ramsey Numbers through different means, and I kind of fell in love with Kalbfleisch's Construction of Special Edge-Chromatic Graphs (1965). One of the gems in here is the way she constructs her graphs, for example, her proof for L(4,4) = 17 comes from constructing a regular graph with red lines of distance 1, 2, 4, and 8, powers of 2 (the blue lines are 3, 5, 6 and 7 but that can also be expressed as 3, 5, 7 and 11 - 4 consecutive primes). Her proof for L(4,5) = 24 colours the graph with red lines of distance 1, 2, 4, 8 and 9, not the clean powers of 2 as above but still quite nice.
I've written some code that explores regular graphs and it's reproduced these findings (validating the code more than anything), and I was testing some combinations for R(5, 5). It turns out I couldn't find a regular 5-clique-free graph of size 38, 39 or 40, but my code got a result for size 41. Here's the colourings: Colour the 1-, 2-, 3-, 5-, 7-, 10-, 13-, 15-, 16-, 17- lines red, and the rest blue. Note we can construct this by taking the first 10 powers of 13 in GF(41) (jumps of 24, 25, 26 and 38 are equivalent to 17, 16, 15 and 3 respectively).
So my two questions are:
- What is a good way to double check my work? I have some level of confidence in my code but I do want to make sure I'm not going down the wrong rabbit hole
- Who else is working in this area? Does anyone see any useful jump from here to improving anything about the current bounds? (the obvious next step would be trying to construct a 5-clique-free graph of n=42 and see if one of Radziszowski's 656 graphs.