Probabilistic Methods Algorithmic Aspects

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Probabilistic Methods Algorithmic Aspects
Prasad Tetali
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Theorem 3. 4 For a finite set V = {vi, . . . , Vn} Q 3?'' with n 5^i;. -0 and \\v, \\ 1. \ 0}. Then we can find in 0(n(P) steps another nontrivial linear dependence a such that for E = {v : a{v) > ^} u* G f: and \E\ n.
Further calculation shows that R{k, k) > ^(1 + o(l))A;2''/^, yielding only a small constant factor improvement. The dependencies being rare, things are brighter when we apply the lemma to the off-diagonal Ramsey numbers R(k, t), with one of the parameters small. We can prove, usi
...ng the lemma, that R{3, t)>{^-o{i))ty\ogH and that There is no known better bound for i?(4, t) without using the local lemma. And this still leaves a gap, since the upper bound is i?(4, .
We conclude this section with yet another application of the local lemma. (Strictly speaking, this is a special case of Theorem 4. 1. ) Consider n points and n sets made out of these n points. Suppose, further, that each set has exactly 10 points and that each point is in exactly 10 sets; i. E. The degree of each set and the degree of each point equal 10.


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