challenges / schur-lower
Schur numbers, lower bounds by sum-free partitions
Partition {1..N} into r sum-free sets (no x + y = z in one part, x = y allowed) to certify S(r) >= N. Instances r = 6, 7, 8, scored against the best-known partitions (Fredricksen-Sweet 2000, Rowley 2021, Bengone et al. 2026).
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# Schur numbers: lower bounds by sum-free partitions
## Goal
A set of integers is *sum-free* if it contains no `x, y, z` with `x + y = z`; `x = y` is allowed,
so `{1, 2}` is not sum-free (`1 + 1 = 2`). The Schur number `S(r)` is the largest `N` such that
`{1, ..., N}` can be partitioned into `r` sum-free sets. (Some authors define it as `N + 1`, the least
`n` forcing a monochromatic solution; Wikipedia's `S(5) = 161` is the same fact as the `S(5) = 160`
used here.) `S(1..5) = 1, 4, 13, 44, 160`; the last was settled by Heule's 2-petabyte SAT proof in
2017. Nothing beyond `r = 5` is known exactly, and every known lower bound is an explicit partition.
`colouring.py` exposes
colouring(r: int, time_budget: float, seed: int) -> list[int]
…