National Heritage Institute

The error of the Guy-Kelly heuristic, measured against exact counts

The Guy-Kelly conjecture predicts that the maximum number of points in general position in an n×n grid is asymptotically c·n. We measure how far the heuristic behind it stands from the truth, using the exact counts of A000755 rather than the threshold it predicts, since a thresho...

Nineteen certified configurations, eighteen independent bounds for the no-four-coplanar problem in the cube, and what they do not establish (OEIS A280537)

This version exists to correct a false novelty claim. Versions up to 2.9 stated that published data for A280537 stop at a(8) = 20. In fact the OEIS entry itself has carried, since January 2017, lower bounds up to a(17) ≥ 42, and Al Zimmermann's 2016 contest Non-Coplanar Points ra...

Balanced orbit defects in half-turn-symmetric no-three-in-line configurations, with new 2n-point configurations for n = 36, 37, 39

A 2n-point no-three-in-line configuration on the n×n grid with half-turn symmetry decomposes into orbits of a larger subgroup of the dihedral group that it happens to contain, together with half-turn pairs that are not such orbits — its orbit defect. Balance lemma. Read as arcs o...

Extremal no-three-in-line subsets of a modular hyperbola in the Hall-Jackson-Sudbery-Wild window

Let p be an odd prime and H_c = {(x,y): xy ≡ c (mod p)} the modular hyperbola in the 2p×2p window of Hall, Jackson, Sudbery and Wild (1975). HJSW observed that keeping three of the four points of each residue class gives 3(p−1) points in general position; Kovács, Nagy and Szabó s...

Extremal no-three-in-line subsets of a modular hyperbola in the Hall-Jackson-Sudbery-Wild window

Let p be an odd prime and H_c = {(x,y): xy ≡ c (mod p)} the modular hyperbola in the 2p×2p window of Hall, Jackson, Sudbery and Wild (1975). HJSW observed that keeping three of the four points of each residue class gives 3(p−1) points in general position; Kovács, Nagy and Szabó s...

The error of the Guy-Kelly heuristic, measured against exact counts

The Guy-Kelly conjecture predicts that the maximum number of points in general position in an n×n grid is asymptotically c·n. We measure how far the heuristic behind it stands from the truth, using the exact counts of A000755 rather than the threshold it predicts, since a thresho...

No four coplanar in the cube: certified witnesses against the prior art this note initially missed (OEIS A280537)

This version exists to correct a false novelty claim. Versions up to 2.9 stated that published data for A280537 stop at a(8) = 20. In fact the OEIS entry itself has carried, since January 2017, lower bounds up to a(17) ≥ 42, and Al Zimmermann's 2016 contest Non-Coplanar Points ra...

Balanced orbit defects in half-turn-symmetric no-three-in-line configurations, with new 2n-point configurations for n = 36, 37, 39

A 2n-point no-three-in-line configuration on the n×n grid with half-turn symmetry decomposes into orbits of a larger subgroup of the dihedral group that it happens to contain, together with half-turn pairs that are not such orbits — its orbit defect. Balance lemma. Read as arcs o...

Balanced orbit defects in half-turn-symmetric no-three-in-line configurations, with new 2n-point configurations for n = 36, 37, 39

A 2n-point no-three-in-line configuration on the n×n grid with half-turn symmetry decomposes into orbits of a larger subgroup of the dihedral group that it happens to contain, together with half-turn pairs that are not such orbits — its orbit defect. Balance lemma. Read as arcs o...

Nineteen certified configurations, eighteen independent bounds for the no-four-coplanar problem in the cube, and what they do not establish (OEIS A280537)

This version exists to correct a false novelty claim. Versions up to 2.9 stated that published data for A280537 stop at a(8) = 20. In fact the OEIS entry itself has carried, since January 2017, lower bounds up to a(17) ≥ 42, and Al Zimmermann's 2016 contest Non-Coplanar Points ra...

Balanced orbit defects in half-turn-symmetric no-three-in-line configurations, with new 2n-point configurations for n = 36, 37, 39

A 2n-point no-three-in-line configuration on the n×n grid with half-turn symmetry decomposes into orbits of a larger subgroup of the dihedral group that it happens to contain, together with half-turn pairs that are not such orbits — its orbit defect. Balance lemma. Read as arcs o...

Nineteen certified configurations, eighteen independent bounds for the no-four-coplanar problem in the cube, and what they do not establish (OEIS A280537)

This version exists to correct a false novelty claim. Versions up to 2.9 stated that published data for A280537 stop at a(8) = 20. In fact the OEIS entry itself has carried, since January 2017, lower bounds up to a(17) ≥ 42, and Al Zimmermann's 2016 contest Non-Coplanar Points ra...

Extremal no-three-in-line subsets of a modular hyperbola in the Hall-Jackson-Sudbery-Wild window

Let p be an odd prime and H_c = {(x,y): xy ≡ c (mod p)} the modular hyperbola in the 2p×2p window of Hall, Jackson, Sudbery and Wild (1975). HJSW observed that keeping three of the four points of each residue class gives 3(p−1) points in general position; Kovács, Nagy and Szabó s...

No four coplanar in the cube: nineteen certified configurations for n <= 29, and what they do not establish (OEIS A280537)

a(n) is the maximum number of points in the n×n×n grid no four of which are coplanar — OEIS A280537. The published data stop at a(8) = 20. This version extends the previous one (n ≤ 7) to n ≤ 29. We exhibit nineteen configurations, each verified by two programs sharing no code an...