Preface; Notation and Conventions; Appetizer: triangles and equations; 1. Forbidding a subgraph; 2. Graph regularity method; 3. Pseudorandom graphs; 4. Graph limits; 5. Graph homomorphism inequalities; 6. Forbidding 3-term arithmetic progressions; 7. Structure of set addition; 8. Sum-product problem; 9. Progressions in sparse pseudorandom sets; References; Index.
An introductory text covering classical and modern developments in graph theory and additive combinatorics, based on Zhao's MIT course.
Yufei Zhao is Associate Professor of Mathematics at the Massachusetts Institute of Technology. His research tackles a broad range of problems in discrete mathematics, including extremal, probabilistic, and additive combinatorics, graph theory, and discrete geometry, as well as applications to computer science. His honors include the SIAM Dénes Kőnig prize (2018), the Sloan Research Fellowship (2019), and the NSF CAREER Award (2021). This book is based on an MIT graduate course, which he has taught and developed over the last five years.
'Yufei Zhao does great mathematics and has an uncanny ability to
explain the deepest results with clear understandable prose. For
anyone interested in the seminal ideas (and their
interrelationships) of recent decades - pseudorandomness, graphons,
graph regularity, to name a few - this is the book to read and
savor.' Joel Spencer, New York University
'This impeccable book should quickly become a classic text in
discrete maths. A huge selection of topics is treated elegantly,
with beautiful illustrations, and in just the `right' amount of
detail to arouse the interest of the reader and leave them well
placed to find out more. In particular, the second half of the book
is a superb introduction to additive combinatorics, which I will
happily recommend to any student in this area.' Ben Green, Oxford
University
'This charming text gives an accessible introduction to the
connected topics of extremal graph theory and modern additive
combinatorics. The focus is very strongly on presenting intuition
and restricting attention to the simplest possible instances of
methods or classes of results, rather than aiming for maximal
generality or the strongest statements; instead, references are
given for further reading, or for the proofs of important theorems
that are only stated here. Being highly suitable for advanced
undergraduates or beginning graduate students, it fills a niche
that is currently not occupied by other texts in these highly
active areas of current mathematical research.' Terry Tao,
University of California, Los Angeles
'A valuable and readable unified treatment of a fast-moving area of
combinatorics from one of the world's experts - sure to become a
standard resource.' Jordan Ellenberg, University of
Wisconsin-Madison
'Yufei Zhao's book is a wonderful book about graph theory, additive
combinatorics, and their surprising connections, involving a major
theme of modern mathematics: the interplay between structure and
randomness. In both areas, the book can take the curious reader,
whether an advanced undergraduate or a professional mathematician,
on a joyous journey from the very basics to state-of-the-art
research. Yufei Zhao himself is a major player in modern research
in both these areas and his presentation is a tour de force.' Gil
Kalai, Hebrew University of Jerusalem and Reichman University
'This is a beautiful treatment of extremal graph theory and
additive combinatorics, focusing on the fruitful interplay between
the two. The book covers the classical results as well as recent
developments in this active area. It is a fascinating manuscript
that would appeal to students and researchers with an interest in
discrete mathematics, theoretical computer number theory, and
related areas.' Noga Alon, Princeton University
'This is a wonderful, well-written account of additive
combinatorics from the graph theoretic perspective. Zhao skillfully
ties in this approach to the usual statements and gives a thorough
development of the subject. This book is indispensable for any
serious researcher in this area. Beginners will find a thorough
account of the subject with plenty of motivation. The more
experienced reader will appreciate the authors' insights and
elegant development of some difficult ideas.' Andrew Granville,
University of Montréal
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