Borsuk's Conjecture Fails in Dimensions 321 and 322
نویسنده
چکیده
n0 = min{n ∈ N | f(n) > n + 1}, then the proof of Kahn and Kalai gives n0 ≤ 1325. On the other hand we know only that n0 ≥ 4 (Perkal [7]; Eggleston [3]). It is of interest where n0 lies. The upper bound on n0 was improved to n0 ≤ 946 (Nilli [6]), n0 ≤ 561 (Raigorodski [8]), n0 ≤ 560 (Weißbach [9]), and n0 ≤ 323 (Hinrichs [4]). In fact, we know that f(n) > n + 1 for all n ≥ 323. Here we show that n0 ≤ 321. Theorem 1 f(321) ≥ 333. Thus, Borsuk’s conjecture fails in all dimensions n ≥ 321.
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