Vinh university college of education mathematics department
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- THE PROBABILITY METHOD IN EXTREMAL FINITE SET THEORY Student: Phạm Tuấn Anh Student code: 205714020910005 Class: 61 th
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VINH UNIVERSITY COLLEGE OF EDUCATION MATHEMATICS DEPARTMENT ---------------------- ESSAY SUBJECT: INTRODUCTION TO PROBABILISTIC METHOD Course: Nhập môn phương pháp xác suất TOPIC: SYSTEM OF EXERCISES APPLYING THE PROBABILITY METHOD IN EXTREMAL FINITE SET THEORY Student: Phạm Tuấn Anh Student code: 205714020910005 Class: 61th High Quality Mathematics Pedagogy Teacher: Dr. Dương Xuân Giáp Nghe An, 2023 PREFACEThere are many known applications of the Probabilistic Method in Extremal Finite Set Theory. In this paper we describe several examples, demonstrating some of the techniques used and illustrating some of the typical results obtained. This is partly a survey paper, but it also contains various new results. The Probabilistic Method is a powerful tool in tackling many problems in Combinatorics. Roughly speaking, the method works as follows: Trying to prove that a combinatorial structure (or a substructure of a given one) with certain desired properties exists, one defines an appropriate probability space of structures and then shows that the desired properties hold in this space with positive probability. Extremal Finite Set Theory is one of the most rapidly developing areas in Combinatorics, which has applications in various other branches of Mathematics and Computer Science including Discrete Geometry, Functional Analysis, Probability Theory and Circuit Complexity. There are numerous known applications of probabilistic arguments in Extremal Finite Set Theory. These include many examples of probabilistic proofs of existence of families of sets with certain properties, as well as proofs of results whose statement often does not seem to suggest any connection to probability. In this paper we describe examples of both types. It is worth mentioning that our choice of the examples described here is limited by the length of this paper and obviously reflects a personal choice as well. It is impossible to even mention all the beautiful known probabilistic proofs in the are. Here we first present simple known probabilistic proofs of three basic results, and then describe several examples of random constructions supplying proofs of existence of certain set systems. Afterwards we describe a few more recent results, and conclude with a section containing various applications of the properties of the entropy function of a random variable to Extremal Finite Set Theory. tải về 120.21 Kb. Chia sẻ với bạn bè của bạn:
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