Chapter 0: Why? What? How? The Probabilistic Method Summer 2020 Freie UniversitΓ€t Berlin
Chapter Outline Why should we learn What is the How will we be the probabilistic probabilistic method? learning this method? semester?
Β§1 Why? Chapter 0: Why? What? How? The Probabilistic Method
Existential Problems Most important questions are existential in nature Theorem 0.1.1 (Mantel, 1907) π 2 The largest triangle-free graph on π vertices has edges. 4 π 2 Lower bound: there is an π -vertex triangle-free graph with edges 4 Upper bound: there is a triangle in any larger graph
Existential Problems - II 1 Riemann Is there some π‘ β β β β2β with ππ π‘ β 2 for which Hypothesis π π‘ = 0 ? Medicine Is there a vaccine for the coronavirus? Does Game of Thronesβ final season have any redeeming Television qualities?
Types of Solutions Constructive solutions are ideal β’ Not only prove the existence of the desired object, but show how to find it β’ Especially important in real-world problems Existential solutions still valuable β’ Prove existence of the object without showing how to find it β’ e.g.: in Mantel upper bound, donβt need to know how to find triangles in dense graphs to show that the complete balanced bipartite graph is optimal Probabilistic method very powerful for proving existential results β’ Often yield randomised constructions as well
Β§2 What? Chapter 0: Why? What? How? The Probabilistic Method
Formulating the Problem Given β’ A (finite) set π» of objects β’ A desired property π¬ Goal β’ Show there is some π β π» with the property π¬ Ramsey Theory β’ π» : all graphs on π vertices β’ π¬ : not having a clique or independent set on π vertices
A Probabilistic Approach Idea β’ Show that a random element π β Ξ© could have the property π¬ Formalism β’ Define a positive probability measure β on the space (Ξ©, 2 Ξ© ) β’ Existence question is equivalent to showing β π β π¬ > 0 Ramsey Theory β’ Take β to be the uniform measure 1 β’ Equivalent to each edge appearing independently with probability 2 π , then the probability of not having a clique or β’ Can show that if π β 2 independent set of size π is positive
Arenβt we just counting? Ramsey Theory β’ Total number of graphs: β’ Number with a clique or independent set of size π : π , this is less than the total number of graphs β’ When π β 2 Advantages of thinking probabilistically β’ Wide array of more advanced tools and methods β’ Will introduce some of these in this course
Β§3 How? Chapter 0: Why? What? How? The Probabilistic Method
Cast and Crew Lectures Exercises Shagnik Das Tibor SzabΓ³ Ander Lamaison Patrick Morris shagnik@mi.fu-berlin.de szabo@math.fu-berlin.de lamaison@math.fu-berlin.de pm0041@math.fu-berlin.de
Online Resources Whiteboard site β’ Make sure you are enrolled at https://mycampus.imp.fu-berlin.de/portal/site/aed0cf99-5d64-48b7-bdb4- 7f5764903675 Course website β’ Further information available at http://discretemath.imp.fu-berlin.de/DMIII-2020/ Cisco Webex Meetings β’ All course events will take place through Webex Meetings β’ Links to the meetings can be found in Whiteboard
Schedule and Lectures Timings β’ Tuesdays and Thursdays, 10:30 β 12:00 β’ Every second week: Thursday lecture replaced by exercise class Lectures β’ Slides available on website in advance β’ Annotated versions available afterwards β’ Please share video, but keep audio muted unless speaking β’ Can also use chat to ask questions β’ Occasional quizzes to check that everything is clear
Homework and Exercise Sessions Problem sheets β’ Posted online every two weeks, usually 1-2 weeks before due date β’ Submit solutions (individually) to the Whiteboard site β’ Can be typed, or photos/scans of (neat) handwritten solutions β’ Indicate which problems should be graded Exercise sessions β’ Will be split into two groups for the exercise sessions β’ During the sessions, you will present and discuss solutions β’ Should sign up on the course website to present a problem β’ When presenting, can show your solution by sharing screen
Grading Aktive Teilnahme β’ Obtain β₯ 60% of possible homework points β’ Present β₯ 3 solutions β’ Be an active participant in exercise classes Exams β’ Final grade comes from an oral exam β’ To be offered in the summer β scheduled later
Any questions?
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