By Bradley Jackson, Dmitri Thoro

ISBN-10: 0201129086

ISBN-13: 9780201129083

Booklet via Jackson, Bradley, Thoro, Dmitri

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Theory Ser. B, 23 (1977), pp. 68-74. The Chips game is in: J. SPENCER, Balancing vectors in the max norm, Combinatorica, 6 (1986), pp. 55-65. and R. J. ANDERSON, L. LOVASZ, P. SHOR, J. SPENCER, E. TARDOS, S. WINOGRAD, Disks, balls and walls: Analysis of a Combinatorial game, Amer. Math. Monthly, 96 (1989), pp. 481-493. The large deviation results are given in an appendix to The Probabilistic Method, referenced in the Preface. This page intentionally left blank LECTURE 5 Discrepancy I Hereditary and linear discrepancy.

I feel that disc(j^) may be small "by accident," whereas herdisc (jtf) and lindisc (sd} give more intrinsic properties of the set-system sd. To make an analogy, a graph G may "accidentally" have o>(G) = ^(G), but if (o(H) = \(H) for all subgraphs H of G, then G is perfect. (3) Let £l = [n] and let M consist of all intervals [1,7], \*
*

For 1 < i < n define a function X, with domain the set of graphs on [n] by That is, Xt(H) is the predicted value of #(//) when we know H on [i] and all other edges are in H with independent probability \. At the extremes X l ( H ) = £[*(G)] is a constant independent of H and Xn(H)=x(H). ) . " Considering H in the probability space of random graphs the sequence X ^ , • • • , Xn forms a martingale, part of a general class known as a Doob Martingale Process. Basically, given that with [i] uncovered the predicted value of x(H) is X,, then the average value of the value predicted when / + 1 is uncovered is still X,.

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