By Claude Flament

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17) follows. c G. R. 4 Hoeffding inequality Let (Yn , Fn ), n ≥ 0, be a martingale. One can obtain bounds for the tail of Y n in terms of the sizes of the martingale differences Dk = Yk − Yk−1 . These bounds are surprisingly tight, and they have had substantial impact in various areas of application, especially those with a combinatorial structure. We describe such a bound in this section for the case when the Dk are bounded random variables. 18) Theorem. Hoeffding inequality. ) for all k and some real sequence (K k ).

6. One-dimensional percolation. Each edge of the one-dimensional lattice Z is declared open with probability p. For k ∈ Z, let r (k) = max{u : k ↔ k + u}, and L n = max{r (k) : 1 ≤ k ≤ n}. Show that P p (L n > u) ≤ np u , and deduce that, for > 0, Pp Ln > (1 + ) log n log(1/ p) →0 as n → ∞. 7. (continuation) Show that, for > 0, Pp Ln < (1 − ) log n log(1/ p) →0 as n → ∞. By suitable refinements of the error estimates above, show that Pp (1 + ) log n (1 − ) log n < Ln < , for all but finitely many n log(1/ p) log(1/ p) = 1.

Let N ≥ 1, let E be a finite set with |E| = N , and let A be a subset of = {0, 1} E with φ( A) ∈ (0, 1). 27) e∈E I A (e) ≥ cφ( A)(1 − φ( A)) log[1/ max I A (e)], e where the reference measure is φ = φ 1 . 28) I A (e) ≥ cφ( A)(1 − φ( A)) Note that φ( A)(1 − φ( A)) ≥ 1 2 log N . N min{φ( A), 1 − φ( A)}. 28). We may assume that m = maxe I A (e) satisfies m > 0, since otherwise φ( A)(1 − φ( A)) = 0. 27) that cφ( A)(1 − φ( A)) m ≥ . 28) follows with an amended value of c, by the monotonicity of m/ log(1/m) or otherwise1 .

### Applications of graph theory to group structure by Claude Flament

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