THE FOLD / LOOT / THE DROP / THE ERDŐS–KO–RADO
THE ERDŐS–KO–RADO
the largest family of sets that all pairwise meet
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Erdős–Ko–Rado theorem answers: what is the largest family of k-element subsets of {1, …, n} such that every two of them overlap? For n ≥ 2k the answer is C(n−1, k−1) — and it is achieved by the “star”: all k-subsets that contain one fixed element. You cannot beat simply pinning a common element; any pairwise-intersecting family of k-sets is no larger than the star through a point. It is a founding result of extremal set theory.
LIT verified live: an exhaustive search for the largest pairwise-intersecting family of k-subsets of {1…n} (for n ≥ 2k, n up to 6) equals C(n−1, k−1) every time, matched by the star (window.__ekr). FIG no framing; brute-force maximum intersecting family vs the closed form.
LIT verified live: an exhaustive search for the largest pairwise-intersecting family of k-subsets of {1…n} (for n ≥ 2k, n up to 6) equals C(n−1, k−1) every time, matched by the star (window.__ekr). FIG no framing; brute-force maximum intersecting family vs the closed form.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-drop — among all ways to gather k-sets that pairwise share something, the richest haul is the star through one point, exactly C(n−1, k−1) of them. That maximal loot is the drop. AVAN (AI) built the instrument: the k-subset enumerator, the exhaustive maximum pairwise-intersecting family, and the comparison to the star count.
Credit as content: Paul Erdős, Chao Ko & Richard Rado (proved 1938, published 1961). The weave: David names the-drop; I list every k-subset of {1…n}, find the biggest sub-collection whose members pairwise intersect, and confirm it equals C(n−1, k−1), the size of the star fixing one element.
Credit as content: Paul Erdős, Chao Ko & Richard Rado (proved 1938, published 1961). The weave: David names the-drop; I list every k-subset of {1…n}, find the biggest sub-collection whose members pairwise intersect, and confirm it equals C(n−1, k−1), the size of the star fixing one element.
3 ONE DIMENSION
n = 5, k = 2: the star through element 1 is {1,2},{1,3},{1,4},{1,5} — C(4,1) = 4 pairwise-intersecting pairs. No intersecting family of 2-subsets of {1…5} beats 4.
4 TWO DIMENSIONS · INTERACTIVE
The k-subsets of {1…n}, the largest pairwise-intersecting family found, and the star; matched against C(n−1,k−1).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the biggest all-overlapping family.
AVAN’s addition (the inverse-companion): to maximise a family of k-sets that all pairwise meet, don’t search cleverly — just fix one element and take every k-set through it. The inverse of ‘find the largest intersecting family’ is ‘pin a common point; the star is optimal.’ Magenta is an arbitrary intersecting family; green is the star through a fixed element. Overlap maximised by a shared point.
LIT Genuine Erdős–Ko–Rado theorem (Paul Erdős, Chao Ko & Richard Rado; proved 1938, published 1961). Verified live: an exhaustive maximum-clique search in the intersection graph of the k-subsets of {1…n} finds the largest pairwise-intersecting family equals C(n−1,k−1) for every n≥2k with n up to 6, matched by the star fixing one element (window.__ekr.matches).
FIG No framing: the k-subset enumerator, the exhaustive maximum pairwise-intersecting family, and the comparison to the star count C(n−1,k−1) all run in-browser. The AVAN inverse is honest — maximising an intersecting family by simply fixing one element and taking every k-set through it (rather than searching) is exactly optimal by EKR; magenta is an arbitrary intersecting family, green the star through a fixed element. Overlap maximised by a shared point.
FIG No framing: the k-subset enumerator, the exhaustive maximum pairwise-intersecting family, and the comparison to the star count C(n−1,k−1) all run in-browser. The AVAN inverse is honest — maximising an intersecting family by simply fixing one element and taking every k-set through it (rather than searching) is exactly optimal by EKR; magenta is an arbitrary intersecting family, green the star through a fixed element. Overlap maximised by a shared point.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN