By Jiwei D., Lee P.K.

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Be independent Bernoulli random variables, with P(Xn = −1) = P(Xn = 1) = 12 for every n, and set Sn √= ni=1 Xi . The law of the iterated logarithm states that lim supn→∞ Sn / 2n log log n = 1 almost surely. Putting it another way, for t ∈ [0, 1], let t = 0. 1 (t) 2 (t) . . be its dyadic expansion, or equivalently, set n (t) = 0 or 1 according as the integer part of 2n t is even or odd. ) Set fn (t) = nk=1 k (t) − n2 . Then the law of iterated logarithm states that fn (t) =1 lim sup n 1/2 n→∞ ( 2 log log n) for almost every t ∈ (0, 1].

Number of edges) of a graph G. Given graphs G and H, the expression H ⊂ G means that H is a subgraph of G. Let F be a ﬁxed graph, usually called the forbidden graph. Set ex(n; F ) = max{e(G) : |G| = n and F ⊂ G}. Paul Erd˝ os: Life and Work 27 and EX(n; F ) = {G : |G| = n, e(G) = ex(n; F ), and F ⊂ G}. We call ex (n; F ) the extremal function, and EX (n; F ) the set of extremal graphs for the forbidden graph F . Then the basic problem of extremal graph theory is to determine, or at least estimate, ex(n; F ) for a given graph F and, at best, to determine EX(n; F ).

Note that if all we care about is whether two r-sets belong to the same class or not, then for every ordered set X with more than r elements, X (r) has precisely 2r distinct canonical partitions, one for each subset I of [r]. If X is inﬁnite then there is only one canonical partition with ﬁnitely many classes: this is the canonical partition belonging to I = ∅, in which all r-sets belong to the same class. Paul Erd˝ os: Life and Work 33 The Erd˝ os-Rado canonical Ramsey theorem claims that if we partition N(r) into any number of classes then there is always an inﬁnite sequence of integers x1 < x2 < .

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