The problem
Claim: every bounded subset S ⊆ ℝ^n decomposes into n+1 subsets of strictly smaller diameter. True for n ≤ 3 (dimension three took until 1955, via Eggleston/Hadwiger/Perles lineages).
Borsuk's Conjecture
historicReferences
History & significance
A natural intuition from simplices and smooth bodies; decades of low-dimensional confirmation bred confidence. Jeff Kahn and Gil Kalai shattered it in 1993 with a counterexample built from the Frankl–Wilson set-system machinery: sets in dimension 1325 requiring exponentially MORE than n+1 parts — later driven down to dimension 65 and even n = 64 by subsequent refinements (Jenrich, Brouwer–Schrijver lineages).
The resolution (human proof)
Disproved 1993 (Kahn–Kalai). The disproof is a founding exhibit of the extremal-set-theory revolution: algebraic-combinatorial methods (mod-p intersection theorems) producing geometric counterexamples beyond geometric imagination. Borsuk's conjecture now lives on honestly labelled: a theorem of low dimensions, a mirage beyond.