|Jan11-10, 05:59 PM||#1|
Ramsey's theorem help..~
1. The problem statement, all variables and given/known data
Ramsey's theorem. Let G be a graph . A clique in G is a subgraph in which every two nodes are connected by an edge . An anti-clique , also called an independent set , is a subgraph in which every two nodes are not connected by an edge . Show that every graph with n nodes contains either a clique or an anti-clique with at least 1/2log2n nodes.
2. Relevant equations
[b]3. The attempt at a solution[/b
Make space for two piles of nodes , A and B . Then , starting with the entire graph , repeatedly add each remaining node x to A if its degree is greater than one half the number of remaining nodes and to B otherwise , and discard all nodes to which x isn't(is) connected if it was added to A(B) . Continue until no nodes are left. At most half of the nodes are discarded at each of these steps , so at least log2 n steps will occur before the process terminates . Each step adds a node to one of the piles , so one of the piles ends up with at least 1/2log2 n nodes. The pile contains the nodes of a clique and the B pile contains the nodes of an anti-clique.
I cant interpret this solution ..~ can anyone help me out..~ pls...
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