Download Combinatorial and geometric group theory: Edinburgh, 1993 by Andrew J. Duncan, N. D. Gilbert, James Howie PDF

By Andrew J. Duncan, N. D. Gilbert, James Howie

The papers during this booklet signify the present kingdom of data in workforce conception. It comprises articles of present curiosity written via such students as S.M. Gersten, R.I. Grigorchuk, P.H. Kropholler, A. Lubotsky, A.A. Razborov and E. Zelmanov. The contributed articles, all refereed, disguise quite a lot of subject matters in combinatorial and geometric crew conception. the quantity may be imperative to all researchers within the quarter.

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Rado, who was an exceptionally tactful man, called the Christian in his problem a 'man', so the problem ran under the title 'Lion and Man'. Years after Besicovitch had found the solution above, Croft investigated several variants of the problem. For example, he proved that if the man has to run along a curve of uniformly bounded curvature then he will be eaten by a clever lion, but not by one who stays on the radius 0 M. Also, n lions can catch a man in an n-dimensional ball, but n - 1 cannot. I believe that the following problem mentioned by Littlewood is still unsolved.

Figure 11 shows eight unit squares touching another. Can we have nine such squares? 65. Infection with Three Neighbours. As in Problem 34, there is an epidemic on the n x n grid, but this time the sites are more resistant: a site becomes infected if at least three (rather than two) of its neighbours are infected. For n = 6k + 2, at least how many sites were infected at the start of an epidemic if eventually all n 2 sites are infected? Fig. 11. A square touching eight others. 16 1. The Problems 66.

7 : c;, 105. This is immediate from Bollobas's Lemma in Problem 104. 106. Assuming, as we may, that the Xk are positive reals, rewrite the bound Lk af = Lk(Ak/ k)P using telescopic sums. 108. A square cannot be cut into an odd number of triangles of equal area. Rather surprisingly, this can be shown with the aid of 2-adic norms. 109. The answer is the minimal t such that t ) n _:: : ( Lt /2J · In particular, the minimum is about log 2 n. 110. Recall from Problem 104 that a collection {(A j, B j) : 1 _:::: j _:::: n} of pairs of sets is cross-intersecting if A j n B j = 0 for every j and A j n Bh "I= 0 whenever j "I= h.

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