Lovasz local lemma graph coloring pages

We prove the asymmetric version of the lemma, from which the symmetric version can be derived. By using the principle of mathematical induction we prove that for all A{displaystyle A} in A{displaystyle {mathcal {A}}} and all subsets S{displaystyle S} of A{displaystyle {mathcal {A}}} that do not include A{displaystyle A}, Pr(A∣⋀B∈SB¯)≤x(A){displaystyle Pr left(Amid bigwedge _{Bin S}{overline {B}}right)leq x(A)}. The induction here is applied on the size (cardinality) of the set S{displaystyle S}. For base case S=∅{displaystyle S=emptyset } the statement obviously holds since Pr(Ai)≤x(Ai){displaystyle Pr(A_{i})leq xleft(A_{i}right)}. We need to show that the inequality holds for any subset of A{displaystyle {mathcal {A}}} of a certain cardinality given that it holds for all subsets of a lower cardinality.

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