So I got my hands on a physics-based constraint solver (think simulated annealing on st eroids) and decided to throw the Ramsey number R(5,5,5) problem at it.
What that means in human terms: Picture a party with 52 people where everyone shakes hands with everyone else. You have 3 colors of handshake (red, blue, green). Can you assign colors so that no group of 5 p eople all have the same color handshakes among themselves? That's 2,598,960 different 5 -person groups to check.
Turns out the answer is YES, and here's the coloring that works: [link to data]
The absolutely insane numbers:
• 1,326 edges to color • 2,598,960 possible K5 cliques to avoid • Search space: 3^1326 = 10^633 possible colorings • For reference: observable universe has ~10^80 atoms • Took 17 minutes on a GPU-accelerated SAT solver
This pushes the lower bound to R(5,5,5) > 52, meaning you need at least 53 vertices bef ore a monochromatic K5 becomes unavoidable in any 3-coloring.
Most Ramsey results are probabilistic ("a random coloring probably works"), but this is an explicit construction - every single edge's color is specified and verifiable.
TL;DR: Found a needle in a haystack the size of 10^553 universes. The needle exists.