Вот новая статья на эту тему -- http://arxiv.org/abs/1010.0383 Автор пишет: "The history of Borsuk's conjecture is somewhat dramatic. Almost all the specialists in the field of combinatorial geometry strongly believed that the conjecture should be true. Multiple results supporting it have been proved. For example, if Ω has smooth boundary, then one certainly obtains f(Ω) <= d+1. This result is due to H.Hadviger [1945-47], and it seems to show the evidence of the conjecture. However, in 1993 J.Kahn and G.Kalai published a break-through paper, where they constructed a finite set of points in a very high dimension d that could not be decomposed into d+1 subsets of smaller diameter. Now, Borsuk's conjecture is known to be true for d<=3 and false for d>=298. Also, we know that ... [f(d) lies between an exponent of the square root of d and an exponent of d]. Here the lower bound was found by A.M.Raigorodskii and the upper estimate is due to O.Schramm." Мой папа безуспешно пытался доказать гипотезу Борсука в последние годы своей жизни.