By McCoy J. A.

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Hence and so A simple greedy algorithm does somewhat better. Let PI , . . , Pm be any maximal subset of [0, x\d with the property that the sets C + Pi are disjoint. We have seen that C + Pi overlaps C + P if and only if P € 1C + Pt. Hence the sets 1C + Pi must cover [0, x ] d . As each such set has measure p,(2C) = 2dp(C) we must have m > xd2~d / n(C}. As before, all sets C + Pi lie in a cube of side x + 2w, w a constant, so that and so A still further improvement appears in the Probabilistic Lens: Efficient Packing (following Chapter 13).

Furthermore X ~ -Ef-X"] almost always. Let us say X i , . . , Xm are symmetric if for every i ^ j there is an automorphism of the underlying probability space that sends event Ai to event Aj. Examples will appear in the next section. In this instance we write and note that the inner summation is independent of i. We set where i is any fixed index. 5 If E[X] ->• oo and A* = o(E[X]) then X > 0 almost always. Furthermore X ~ E[X] almost always. 5 has the intuitive sense that conditioning on any specific Ai holding does not substantially increase the expected number E[X] of events holding.

We shall bound the failure probability by k(l — p)n + k2p. 1 then assures us that with positive probability the algorithm succeeds. This, by our usual magic, means that there is some running of the algorithm which yields a final coloring with no monochromatic e, that is, there exists a two-coloring of V with no monochromatic edge. For convenience, we bound the probability that some e 6 H is red in the final coloring, the failure probability for the algorithm is at most twice that. An e G E can be red in the final coloring in two ways.

### A Bernstein property of solutions to a class of prescribed affine mean curvature equations by McCoy J. A.

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