# Math 595

Container Method, Spring 2019

**Instructor:** Jozsef Balogh

**Office:** 233B Illini Hall

**E-mail:** jobal@math.uiuc.edu

**Time and place:** 11:00- 11:50 pm MWF, 145 Altgeld Hall

**Office Hour:** by appointment.

PREREQUISITES:

Math 580 or consent of instructor, obtainable by familiarity
with elementary combinatorics.

CANCELLED CLASSES: ;

MAKE UP CLASSES:

;

COURSE REQUIREMENTS:

Attendance + writing typed class notes.

Lecture Notes:

These are class notes, intended for lecturing purposes. The targeted level is at young graduate students; for research applications, I recommend reading the original papers. The history of the
problems are not complete, for those see the cited papers. Also, we do not try to prove the best possible
results, and skip some technical details; the aim is to provide some notes which could be used for teaching purposes.
The notes are topic wise, the length of an average note to teach varies between 30 and 80 minutes.
Feel free to send me comments, as at one point I would like to revise them.

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Lecture 1: Non-technical statement of the Container Lemma, with some applications.**

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Lecture 2-3: Graph container method, number of q-colorings of d-regular graphs (by Galvin).**

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Lecture 4: Random variant of Sperner Theorem (Balogh-Mycroft-Treglown).**

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Lecture 5: Number of Sidon sets (via graph containers).**

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Lecture 6: Number of C4-free graphs (Kleitman-Winston).**

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Lecture 7: Rigorous statement of the Container Lemma (as in [BMS]), with the k-AP application.**

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Lecture 8(i): Turan theorem in random graphs.**

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Lecture 8(ii): Solution of the KLR-conjecture.**

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Lecture 9: Number of maximal triangle-free graphs (Balogh-Petrickova).**

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Lecture 10: Number of maximal sum-free sets (Balogh-Liu-Sharifzadeh-Treglown).**

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Lecture 11: (3,4)-problem (Balogh-Solymosi).**

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Lecture 12: Eps-nets for points and lines (Balogh-Solymosi).**

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Lecture 13-14: Container Lemma for 3-uniform hypergraphs.**

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Lecture 15: Volume computing **

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Lecture 16-17: Folkman numbers (Rodl-Rucinski-Schacht); Random Ramsey (Nenadov-Steger) **

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Lecture 18-19: List coloring **

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Lecture 20: Note on Induced Ramsey Numbers (Conlon Dellamonica, La Fleur, Rodl Schacht) **

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Lecture 21: Ferber-McKinley_Samotij (supersaturation) **

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Lectures 22-23: Saxton-Thomason proof, simple container **

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Lectures 24-25: Anush proof vith logic, container lemma **

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Lectures 26-27: Proof of k-uniform container lemma from BMS. **