Help me understand... if dude proved it was false with 4^10000 exponential graph, why hasn’t a smaller tensor been tested? If he could derive a solution with a large set, shouldn’t it be easy enough to test a half size set and classic sort out where the smallest solution is?
IANA graph theorist, but I don't think it works like that. Sounds like his graph G (with ~4^100 nodes, from which the 4^10000 node exponential graph and the counter-exemplary tensor product itself derive) is a bespoke construction. I don't think it's possible in practice to just drop nodes from it and test whether the condition holds for the new tensor product. There are 4^100 ways to remove r = 1 node from an n = 4^100 node graph. Increase r and the number of combinations you'd need to test shoots through the roof. Without some way to shave off (the vast majority of) candidates, the possible solution space seems way too big for brute force search.
Someone with real expertise, please tell me if I'm wrong.
That's right. That's a general proof technique: to assemble a structure from a bunch of parts or layers, each of which adds to or multiplies the size of the structure, and possibly the various factors may be known only by upper bounds.
So there is room to improve slightly by brute force or more careful accounting, or a lot by finding an alternate construction.
