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extremal graph theory
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Undergraduate
By
Shiva Kintali
on May 31, 2012 | Updated Dec. 6, 2017
Party Problem
Suppose there are six people at a party. Prove that there are always three of them so that every two know each other (or) no two know each other. In other words, let the edges of the complete graph o…
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extremal graph theory
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Undergraduate
By
Shiva Kintali
on June 17, 2012 | Updated Dec. 6, 2017
Turan geometry
Let \(x_1, x_2, \dots, x_n\) be a set of diameter one in the plane. Prove that the maximum number of pairs of points at distance greater than \(1/\sqrt{2}\) is \(\lfloor n^2/3\rfloor\).
Mathematics
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extremal graph theory
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Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Extremal graphs
Let \(G\) be a simple undirected graph with \(2n\) nodes and no triangles (i.e., cycles of length \(3\)). Prove that \(\mathcal G\) has at most \(n^2\) edges.
Mathematics
Graph Theory
extremal graph theory
triangles
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Undergraduate
By
Shiva Kintali
on Oct. 12, 2013 | Updated Jan. 4, 2018
Turan graphs
Prove the following : Let \(G\) be a simple graph with \(n\) vertices. If \(G\) has more than \(\lfloor \frac{n^2}{4} \rfloor\) edges, then \(G\) contains a triangle. Let \(G\) be a simple graph…
Mathematics
Graph Theory
extremal graph theory
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Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Basics of Ramsey's theory
A special case of Ramsey's theorem states that for any pair of positive integers \((r,s)\), there exists a least positive integer \(R(r,s)\) such that for any complete graph on \(R(r,s)\) vertices, wh…
Mathematics
Graph Theory
extremal graph theory
ramsey theory
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