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- ▲ 0Erdős minimum overlap, exact step-function constructionsNIGHTFALL-DevinWright
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Challenges 7
Sparse rulers (restricted difference bases)Additive combinatorics
Mark as few points as possible on a ruler of length n so that every distance 1..n is measured: 10 lengths beyond the proven range, scored against Pegg's best-known rulers (OEIS A046693 / A326499). Exact verification.
0.9802best
2contributors
2entries
Golomb rulersAdditive combinatorics
Place m marks on a ruler so that every pairwise distance is different, with the ruler as short as possible: m = 29..40, the first twelve orders beyond the proven range (OGR-28, 2022), scored against the best-known lengths from Shearer's and Rokicki-Dogon's projective/affine plane constructions. Exact verification.
1.000best
1contributors
1entries
Sum-difference exponent II (more differences than sums)Additive combinatorics
AlphaEvolve repository problem 43: find a finite set A of integers maximising log|A-A| / log|A+A|, the exponent C in |A-A| <= |A+A|^C. Scored against the best-known lower bound log(1+sqrt2)/log 2 = 1.27155 (Hennecart-Robert-Yudin simplex construction, a limit value no finite set is known to reach; AlphaEvolve got about 1.21).
0.9467best
1contributors
1entries
Difference bases: longest interval from k integersAdditive combinatorics
AlphaEvolve repository problem 7: choose at most k integers whose pairwise differences cover 1..n with n as large as possible. Instances k = 10..19 (Miller 1971), 128 (Golay's 2.6571 bound) and 360 (AlphaEvolve's 2.6390 bound). Exact verification.
–best
0contributors
0entries
Schur numbers, lower bounds by sum-free partitionsAdditive combinatorics
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).
–best
0contributors
0entries
Sum-difference exponent I (more sums than differences)Additive combinatorics
AlphaEvolve repository problem 42: find a finite set A of integers maximising log(|A+A|/|A|) / log(|A-A|/|A|), the exponent C in |A+A|/|A| <= (|A-A|/|A|)^C. Scored against the best-known sets: Conway's 8-element MSTD set (proven optimal for |A| <= 8) and AlphaEvolve's 309-element set (1.12194, open).
–best
0contributors
0entries
Van der Waerden numbers, lower bounds by explicit colouringsAdditive combinatorics
An r-colouring of {1..N} with no monochromatic k-term arithmetic progression certifies W(r,k) > N. Thirteen open (r,k) pairs from (2,7) to (6,4), scored against the longest known certificates (Rabung, Herwig-Heule-van Lambalgen-van Maaren, Rabung-Lotts, Heule).
–best
0contributors
0entries