''Extremal Problems For Finite Sets'' is a mathematics book on the

extremal combinatorics Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy ce ...

of finite sets and families of finite sets. It was written by

Péter Frankl Péter Frankl (born 26 March 1953 in Kaposvár, Somogy County, Hungary) is a mathematician, busking, street performer, columnist and educator, active in Japan. Frankl studied Mathematics at Eötvös Loránd University in Budapest and submitted ...

and Norihide Tokushige, and published in 2018 by the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

as volume 86 of their Student Mathematical Library book series. The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has suggested its inclusion in undergraduate mathematics libraries.

Topics

The book has 32 chapters. Its topics include: *

Sperner's theorem Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set theory. It is named after Emanuel Sperner, who ...

, on the largest

antichain In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its wid ...

in the family of subsets of a given finite set. *The Sauer–Shelah lemma, on the largest size of a family of sets that avoids shattering any set of given size. *The

Erdős–Ko–Rado theorem In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it ...

, on the largest pairwise-intersecting family of subsets of a given finite set, with multiple proofs; the closely related

Lubell–Yamamoto–Meshalkin inequality In combinatorial mathematics, the Lubell–Yamamoto–Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by , , , and . It is named for the initials of three of its dis ...

; the Hilton-Milner theorem, on the largest intersecting family with no element in common; and a conjecture of

Václav Chvátal Václav (Vašek) Chvátal () is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada and a Visiting Professor at Charles University in Prague. He has published ext ...

that the largest intersecting family of any downward-closed family of sets is always achieved by a family with an element in common. *The Kruskal–Katona theorem relating the size of a family of equal-sized sets and the size of the family of subsets of its sets of a smaller equal size. *

Cap set In affine geometry, a cap set is a subset of \mathbb_3^n (an n-dimensional affine space over a three-element field) with no three elements in a line. The cap set problem is the problem of finding the size of the largest possible cap set, as a func ...

s and the

sunflower conjecture The common sunflower (''Helianthus annuus'') is a large annual forb of the genus ''Helianthus'' grown as a crop for its edible oily seeds. Apart from cooking oil production, it is also used as livestock forage (as a meal or a silage plant), as ...

on families of sets with equal pairwise intersection. *Open problems including Frankl's

union-closed sets conjecture The union-closed sets conjecture is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said to be ''union-closed'' if the union of any two sets from the family belongs to the family. The conjecture states: For e ...

. Many other results in this area are also included.

Audience and reception

Although the book is intended for undergraduate mathematics students, reviewer Mark Hunacek suggests that readers will either need to be familiar with, or comfortable looking up, terminology for hypergraphs and

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

s. He suggests that the appropriate audience for the book would be advanced undergraduates who have already demonstrated an interest in combinatorics. However, despite the narrowness of this group, he writes that the book will likely be very valuable to them, as the only source for this material that is written at an undergraduate level.

References

{{reflist, refs= {{citation, title=Review of ''Extremal Problems For Finite Sets'', journal=MAA Reviews, publisher=Mathematical Association of America, first=Mark, last=Hunacek, date=October 2018, url=https://www.maa.org/press/maa-reviews/extremal-problems-for-finite-sets {{citation, title=Review of ''Extremal Problems For Finite Sets'', journal=Mathematical Reviews, first=Fred C., last=Holroyd, mr=3822342 {{citation, title=Review of ''Extremal Problems For Finite Sets'', journal=zbMATH, first=M. P., last=Chaudhary, zbl=1416.05001 Combinatorics Families of sets Mathematics books 2018 non-fiction books Publications of the American Mathematical Society