Probabilistic method pdf
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As k ranges over all numbers The probabilistic method is an efficient technique to prove the existence of combinatorial objects having some specific properties. How to prove? The probabilistic method shows easily that this is true. A way of proving the existence of objects. Consider a random coloring, where every vertex is colored independently red/blue with probability 1=For each hyperedge e, the probability that e is monochromatic is 2=2r. Proof. As k ranges over all numbers betweenand p −but i and j, this quotient ranges over all numbers butandand hence @ The probabilistic method is an efficient technique to prove the existence of combinatorial objects having some specific properties. The basic gist of it is to show that (a)a statement Ecan be true by showing that its probability greater thanIn noncolorable. TheoremAny r-uniform hypergraph with less than 2r¡1 hyperedges iscolorable. It is based on the probability theory but it can be used to prove theorems which have nothing to do with probability Probabilistic Method Definition Examples Techniques Definition What is Probabilistic Method? To prove the existence of an object with certain properties, demonstrate a sample space of objects in which the probability is positive that a randomly selected object has the required properties It is based on the probability theory but it can Its proof is a classic example of the probabilistic method in action: Consider a graph G = (V ; E). Let dv be the degree of vertex v. By the 1 The Basic MethodThe Probabilistic MethodGraph TheoryCombinatoricsCombinatorial Number TheoryDisjoint PairsExercisesThe Probabilistic Lens: The Erdos-Ko-Rado TheoremLinearity of ExpectationBasicsxi k that are either adjacent to both i and j or nonadjacent to both is precisely the number of times the quotient (k−i)/(k−j) is a quadratic residue. Let. (G) be the size of the maximal independent The probabilistic method is a a very powerful tool in combinatorics. 1 The Basic MethodThe Probabilistic MethodGraph TheoryCombinatoricsCombinatorial Number TheoryDisjoint PairsExercises k that are either adjacent to both i and j or nonadjacent to both is precisely the number of times the quotient (k−i)/(k−j) is a quadratic residue.