Cliques and Independent Sets prepared and instructed by
Cliques and Independent Sets prepared and instructed by Shmuel Wimer Eng. Faculty, Bar-Ilan University March 2020 Cliques and Independent Sets 1
Turán’s Theorem March 2020 Cliques and Independent Sets 2
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independent March 2020 Cliques and Independent Sets 8
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Application to Combinatorial Geometry 9 good pair, 6 bad pairs. 12 good pair, 3 bad pairs. March 2020 Cliques and Independent Sets 10
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Ramsey's Theorem March 2020 Cliques and Independent Sets 14
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The determination of Ramsey number in general is a very difficult unsolved problem. March 2020 Cliques and Independent Sets 18
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Bounds on Ramsey Numbers March 2020 Cliques and Independent Sets 20
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Shannon Capacity We study the relation between maximum independent set and the maximum number of distinct words that can be safely used in a message over a noisy channel. Some pairs of chars are so similar that they can be confounded by the receiver due to noise What is the largest number of distinct words that can be used in messages without a confusion at the receiver due to noise? March 2020 Cliques and Independent Sets 25
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0 4 0 0 4 4 1 1 ⊗ 3 2 3 2 1 0 0 1 2 3 4 0 The strong product is embedded on a torus. All 0 vertices are identical, mapped to unique point on the torus. March 2020 Cliques and Independent Sets 27
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0 4 3 2 1 0 March 2020 0 Cliques and Independent Sets 1 2 3 4 0 29
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