R(3,3). invite 6 people to a party and 3 of them are mutual friends or 3 are mutual strangers. guaranteed. with 5 people you can dodge it.
the party problem. greenwood and gleason, 1955
tap or click the right half, or press → / space
R(4,4). the same game for groups of 4 needs 18 people. proved in 1955. checking by hand: 2153 colorings of K18's edges.
greenwood and gleason, 1955
R(5,5) is one of these four numbers. nobody on earth knows which. erdos said if aliens demand R(5,5) we compute it; if they demand R(6,6) we attack.
angeltveit and mckay pushed R(5,5) ≤ 46 in 2024
S(5), the 5th schur number. proved by a SAT solver in 2018. the proof certificate is 2 petabytes, the largest proof ever produced.
heule, 2018
my corner of this world: 2-color rado numbers of ax + by = az. my SAT pipeline found they equal a squared, verified past a = 30. small numbers, but ones nobody had written down.
kissat + pysr pipeline · jeffdoesmath.substack.com
the number of ways to 2-color the integers the S(5) proof had to rule out. atoms in the observable universe: about 1080. this has digits.
digit count computed live: floor(160879 · log10 2) + 1
the ceiling of this tour. an upper bound from a ramsey problem. arrow notation: g1 = 3↑↑↑↑3 already cannot fit in the universe. graham's number is g64. the answer might just be 13.
graham and rothschild, 1971. best known lower bound: 13 (barkley, 2008)
how much disorder can you have before order is forced?
every number here came from the same question. that is ramsey theory. i spend my days asking a SAT solver.