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Statement: Prove the Erdōs-Ko-Rado theorem for intersecting families of sets. The theorem asserts that if a family of sets has the property that every pair of sets intersects, then the family is maximized by the family of all sets containing a fixed element. The proof should involve combinatorial techniques, including the use of extremal set theory, and explore the generalization of this theorem to other combinatorial structures. Required Research: 1. "Combinatorial Set Systems and the Erdős-Ko-Rado Theorem" [https://www.jstor.org/stable/43740558] 2. "Erdós-Ko-Rado Theorem and Applications in Combinatorics" [https://www.sciencedirect.com/science/article/abs/pii/S0167488904700003] 3. "Intersecting Families and Their Applications in Combinatorics" [https://www.springer.com/gp/book/9780387945361]

Statement: Prove the Erdōs-Ko-Rado theorem for intersecting families of sets. The theorem asserts that if a family of sets has the property that every pair of sets intersects, then the family is maximized by the family of all sets containing a fixed element. The proof should involve combinatorial techniques, including the use of extremal set theory, and explore the generalization of this theorem to other combinatorial structures. Required Research: 1. "Combinatorial Set Systems and the Erdős-Ko-Rado Theorem" [https://www.jstor.org/stable/43740558] 2. "Erdós-Ko-Rado Theorem and Applications in Combinatorics" [https://www.sciencedirect.com/science/article/abs/pii/S0167488904700003] 3. "Intersecting Families and Their Applications in Combinatorics" [https://www.springer.com/gp/book/9780387945361]

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 10EQ
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Statement: Prove the Erdōs-Ko-Rado theorem for intersecting families of sets. The theorem asserts that if a family of sets has the property that every pair of sets intersects, then the family is maximized by the family of all sets containing a fixed element. The proof should involve combinatorial techniques, including the use of extremal set theory, and explore the generalization of this theorem to other combinatorial structures. Required Research: 1. "Combinatorial Set Systems and the Erdős-Ko-Rado Theorem" [https://www.jstor.org/stable/43740558] 2. "Erdós-Ko-Rado Theorem and Applications in Combinatorics" [https://www.sciencedirect.com/science/article/abs/pii/S0167488904700003] 3. "Intersecting Families and Their Applications in Combinatorics" [https://www.springer.com/gp/book/9780387945361]
Transcribed Image Text:Statement: Prove the Erdōs-Ko-Rado theorem for intersecting families of sets. The theorem asserts that if a family of sets has the property that every pair of sets intersects, then the family is maximized by the family of all sets containing a fixed element. The proof should involve combinatorial techniques, including the use of extremal set theory, and explore the generalization of this theorem to other combinatorial structures. Required Research: 1. "Combinatorial Set Systems and the Erdős-Ko-Rado Theorem" [https://www.jstor.org/stable/43740558] 2. "Erdós-Ko-Rado Theorem and Applications in Combinatorics" [https://www.sciencedirect.com/science/article/abs/pii/S0167488904700003] 3. "Intersecting Families and Their Applications in Combinatorics" [https://www.springer.com/gp/book/9780387945361]
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