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Area of combinatorics in mathematics
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are inverse problems: given the size
Additive_combinatorics
Mathematical subject
arithmetic combinatorics in his review of "Additive Combinatorics" by Tao and Vu. Szemerédi's theorem is a result in arithmetic combinatorics concerning
Arithmetic_combinatorics
Polish-Canadian mathematician
specialties are harmonic analysis, geometric measure theory, and additive combinatorics. Łaba earned a master's degree in 1986 from the University of Wrocław
Izabella_Łaba
One of several theorems in different areas of mathematics
often called Schur's property, also due to Issai Schur. The Wikibook Combinatorics has a page on the topic of: Proof of Schur's theorem In Ramsey theory
Schur's_theorem
Branch of discrete mathematics
making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph
Combinatorics
Class of norms in additive combinatorics
In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like
Gowers_norm
Study of subsets of integers and behavior under addition
the Erdős–Turán conjecture on additive bases. Shapley–Folkman lemma Additive combinatorics Multiplicative combinatorics Multiplicative number theory Nathanson
Additive_number_theory
Long dense subsets of the integers contain arbitrarily large arithmetic progressions
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured
Szemerédi's_theorem
Sumset of a field subject to a specific polynomial restriction
In additive number theory and combinatorics, a restricted sumset has the form S = { a 1 + ⋯ + a n : a 1 ∈ A 1 , … , a n ∈ A n a n d P ( a 1 , …
Restricted_sumset
Mathematical constant
G ) {\displaystyle D(G)} is an invariant of a group studied in additive combinatorics, quantifying the size of nonunique factorizations. Given a finite
Davenport_constant
British mathematician
Sanders FRS is an English mathematician, working on problems in additive combinatorics at the interface of harmonic analysis and analytic number theory
Tom_Sanders_(mathematician)
In additive combinatorics, the Plünnecke–Ruzsa inequality is an inequality that bounds the size of various sumsets of a set B {\displaystyle B} , given
Plünnecke–Ruzsa_inequality
Influence of local substructure of a graph on global properties
graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence
Extremal_graph_theory
Australian and American mathematician (born 1975)
partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing, and
Terence_Tao
In additive combinatorics, the Ruzsa triangle inequality, also known as the Ruzsa difference triangle inequality to differentiate it from some of its variants
Ruzsa_triangle_inequality
Mathematical subject
Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory. Ergodic Ramsey theory arose shortly
Ergodic_Ramsey_theory
type of numerical sequence playing a role in ergodic theory and additive combinatorics. The concept is related to nilpotent Lie groups and almost periodicity
Nilsequence
Set disjoint from its sumset with itself
In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words
Sum-free_set
Chinese-American mathematician
Euclidean harmonic analysis, analytic number theory, geometry and additive combinatorics. He is an assistant professor in the Department of Mathematics at
Ruixiang_Zhang
Theorem about prime numbers
"A Multidimensional Szemerédi Theorem in the primes via Combinatorics". Annals of Combinatorics. 22 (4): 711–768. arXiv:1306.3025. doi:10.1007/s00026-018-0402-4
Green–Tao_theorem
American-Turkish mathematician (born 1992)
Koymans, Peter; Pagano, Carlo (2024). "Hilbert's tenth problem via additive combinatorics". arXiv:2412.01768 [math.NT]. Burgess, Kaya (July 21, 2026). "Harvard
Levent_Alpöge
On the approximate structure of sets whose sumset is small
In additive combinatorics, a discipline within mathematics, Freiman's theorem is a central result which indicates the approximate structure of sets whose
Freiman's_theorem
uncertainty. Additive combinatorics The part of arithmetic combinatorics devoted to the operations of addition and subtraction. Additive number theory
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Theorem in arithmetic combinatorics on finite partitions of the natural numbers
theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever the natural numbers
Folkman's_theorem
Theorem in arithmetic combinatorics
In arithmetic combinatorics, the Erdős–Szemerédi theorem states that for every finite set A {\displaystyle A} of integers, at least one of the sets A
Erdős–Szemerédi_theorem
One of several related theorems regarding the sizes of certain sumsets in abelian groups
In the branch of mathematics known as additive combinatorics, Kneser's theorem can refer to one of several related theorems regarding the sizes of certain
Kneser's theorem (combinatorics)
Kneser's_theorem_(combinatorics)
British mathematician (born 1977)
Green's research is in the fields of analytic number theory and additive combinatorics, but he also has results in harmonic analysis and in group theory
Ben_Green_(mathematician)
Israeli mathematician
1971) is an Israeli mathematician specializing in Ergodic theory, Additive combinatorics and Number theory. She holds the Henry and Manya Noskwith Chair
Tamar_Ziegler
Progression-free set of numbers
In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression
Salem–Spencer_set
known for his work in the field that would eventually be called Additive Combinatorics. Particularly notable was his "ingenious" application of the Szemerédi–Trotter
György_Elekes
Minimum monochromatic-triangle theorem in graph theory
Combinatorics, Probability and Computing. 31 (5): 907–923. doi:10.1017/S0963548322000074. Zhao, Yufei (2023). Graph Theory and Additive Combinatorics:
Goodman's_theorem
North American undergraduate mathematics award
Sawhney (Combinatorics, Massachusetts Institute of Technology), Cynthia Stoner (Combinatorics, Harvard University), Ashwin Sah (Combinatorics, Massachusetts
Morgan_Prize
Mathematical concept
was introduced in the 2010s but can be traced to older sources in additive combinatorics. Let G {\displaystyle G} be a group and K ≥ 1 {\displaystyle K\geq
Approximate_group
Mathematics professor
He is the author of several important results in combinatorics (especially additive combinatorics), harmonic analysis and other areas. In 2003, jointly
Nets_Katz
Method in combinatorics
constraints. Such questions arise naturally in extremal graph theory, additive combinatorics, discrete geometry, coding theory, and Ramsey theory; they include
Container_method
Mathematical monograph
overview additive combinatorics. Similarly, although Cassels notes the existence of material on additive combinatorics in the books Additive Zahlentheorie
Sequences_(book)
Statement in arithmetic combinatorics
In arithmetic combinatorics, the corners theorem states that for every ε > 0 {\displaystyle \varepsilon >0} , for large enough N {\displaystyle N} , any
Corners_theorem
Theorem in graph theory
(1978), "Triple systems with no six points carrying three triangles", Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math
Graph_removal_lemma
On the existence of arithmetic progressions in subsets of the natural numbers
Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the natural
Roth's theorem on arithmetic progressions
Roth's_theorem_on_arithmetic_progressions
Periodic sequence of integers whose partial sums are triangular numbers
In additive combinatorics, a Šindel sequence is a periodic sequence of integers with the property that its partial sums include all of the triangular numbers
Šindel_sequence
Mathematical conjecture
mathematics Rudin's conjecture is a mathematical conjecture in additive combinatorics and elementary number theory about an upper bound for the number
Rudin's_conjecture
Mathematical inequality relating inner products and norms
p. 71. ISBN 9780387938394. Zhao, Yufei (2023). Graph theory and additive combinatorics: exploring structure and randomness. Cambridge ; New York, NY: Cambridge
Cauchy–Schwarz_inequality
Conjecture in additive combinations about subsets of natural numbers
In additive combinatorics, the Erdős sumset conjecture is a conjecture which states that if a subset A {\displaystyle A} of the natural numbers N {\displaystyle
Erdős_sumset_conjecture
Algebraic structure
ISBN 9783110283600 Green, Ben (2005), "Finite field models in additive combinatorics", Surveys in Combinatorics 2005, Cambridge University Press, pp. 1–28, arXiv:math/0409420
Finite_field
Greek mathematician and information theorist
information-theoretic ideas and results in probability theory and additive combinatorics. Kontoyiannis earned a B.S. in mathematics from Imperial College
Ioannis_Kontoyiannis
On solvability of Diophantine equations
Hilbert's 10th problem is undecidable for every ring of integers using additive combinatorics. Another team of mathematicians subsequently claimed another proof
Hilbert's_tenth_problem
British mathematician
New England). Austin works in ergodic theory, harmonic analysis, additive combinatorics, metric geometry, high-dimensional probability, and rigorous statistical
Tim_Austin_(mathematician)
Length in a vector space
descriptions of redirect targets Gowers norm – Class of norms in additive combinatorics Kadec norm – All infinite-dimensional, separable Banach spaces are
Norm_(mathematics)
Unsolved problem in number theory
conjecture is an old unsolved problem in additive number theory posed by Paul Erdős and Pál Turán in 1941. It concerns additive bases, subsets of natural numbers
Erdős–Turán conjecture on additive bases
Erdős–Turán_conjecture_on_additive_bases
Points with no three in a line
c<3} was considered one of the most intriguing open problems in additive combinatorics and Ramsey theory for over 20 years, highlighted, for instance,
Cap_set
Peruvian mathematician (born 1977)
Mathematical Society "for contributions to analytic number theory, additive combinatorics and combinatorial group theory". "Zentralblatt MATH". Harald Helfgott
Harald_Helfgott
specialist in harmonic analysis, geometric measure theory, and additive combinatorics Carole Lacampagne, American mathematician known for her work in
List_of_women_in_mathematics
Area of discrete mathematics
ISBN 0-89791-785-5. MR 1427525. Zhao, Yufei (2023). Graph Theory and Additive Combinatorics: Exploring Structure and Randomness. Cambridge University Press
Graph_theory
Artificial intelligence method for mathematical discovery
FunSearch was first demonstrated on the cap set problem, a problem in additive combinatorics concerning the largest possible subset of Z 3 n {\displaystyle \mathbb
FunSearch
American mathematician
Salem Prize (joint with Julian Sahasrabudhe) for contributions to additive combinatorics and related fields, including her work on quantitative density theorems
Sarah_Peluse
Existence theorem for economical additive bases of every order
In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive bases of every order
Erdős–Tetali_theorem
Bound on the number of incidences between points and lines in the plane
incidence geometry and the Erdős-Szemerédi sum-product problem in additive combinatorics. We may discard the lines which contain two or fewer of the points
Szemerédi–Trotter_theorem
Mathematical problem about tiling the plane with stripes
I. Z. (2013). "A note on the pyjama problem". European Journal of Combinatorics. 34 (7): 1071–1077. arXiv:1211.6138. doi:10.1016/j.ejc.2013.03.001.
Pyjama_problem
Sequence of numbers avoiding sums of subsets
In mathematics, a sum-free sequence is an increasing sequence of positive integers, a 1 , a 2 , a 3 , … , {\displaystyle a_{1},a_{2},a_{3},\ldots ,} such
Sum-free_sequence
Mathematical function
descriptions of redirect targets Gowers norm – Class of norms in additive combinatorics Locally convex topological vector space – Space with topology generated
Seminorm
Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics: CIRM Jean-Morlet Chair, Fall 2016. Springer. p. 185. ISBN 9783319749082
List_of_conjectures
was awarded the European Prize in Combinatorics for "his profound results in extremal and probabilistic combinatorics particularly for his result on independent
Robert_Morris_(mathematician)
Sums vector sets A and B by adding each vector in A to each vector in B
Cambridge: Cambridge University Press Tao, Terence & Vu, Van (2006), Additive Combinatorics, Cambridge University Press Zelenyuk, V (2015). "Aggregation of
Minkowski_addition
Application of geometry in number theory
been applied to the study of sumsets, one of the key objects of additive combinatorics. Minkowski's geometry of numbers had a profound influence on functional
Geometry_of_numbers
Sparse graph with strong connectivity
algebraic and group-theoretic, the second strategy is analytic and uses additive combinatorics, the third strategy is combinatorial and uses the zig-zag and related
Expander_graph
Indian mathematician (born 1951)
called the Balu-Koblitz Theorem. His work in Additive Combinatorics includes his two page paper on additive complements of squares, hence disproving a long
Ramachandran_Balasubramanian
Lie group of complex numbers of unit modulus; topologically a circle
Springer. ISBN 978-0-387-78214-0. Tao, Terence; Vu, Van H. (2006). Additive Combinatorics. Cambridge Studies in Advanced Mathematics. Vol. 105. Cambridge
Circle_group
Sums of sets of vectors are nearly convex
non-convexities of the summand functions. Ekeland and later authors argued that additive separability produced an approximately convex aggregate problem, even though
Shapley–Folkman_lemma
British mathematician specialising in arithmetic combinatorics
Prize "in recognition of her outstanding contributions to additive number theory, combinatorics and harmonic analysis and to the mathematical community
Julia_Wolf
many edges. Thus it has its application in extremal graph theory, additive combinatorics and Ramsey theory. Let u , n , r , m , t ∈ N {\displaystyle u,n
Dependent_random_choice
Polish mathematician (born 1983)
Sciences at the Tel Aviv University. He is known for his work in combinatorics, additive number theory, Ramsey theory and graph theory. He studied at the
Wojciech_Samotij
Topics referred to by the same term
Davenport constant, in mathematics, an invariant of a group studied in additive combinatorics Davenport diagram, a graphical tool used in acid base physiology
Davenport
Forbidden graph characterization Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Probabilistic Combinatorics, Béla Bollobás, 1986, ISBN 0-521-33703-8
Forbidden_subgraph_problem
Hungarian mathematician
American Mathematical Society, "for contributions to extremal combinatorics, probability and additive number theory, and for graduate mentoring". In 2024 he
József_Balogh_(mathematician)
Surname list
Gowers, FRS (born 1963), British mathematician Gowers norm, in additive combinatorics Trevor Gowers (1945–1996), Australian rules footballer Walter Gowers
Gowers
Cambridge University Press. ISBN 978-0-521-84903-6. Multiplicative combinatorics Additive combinatorics Additive number theory Sum-product phenomenon
Multiplicative_number_theory
non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication
Free_convolution
Vietnamese mathematician
represented as a subsum? In 2006, with Tao and Vu published their book "Additive Combinatorics". Together, they developed the Inverse Littlewood-Offord theory
Van_H._Vu
Differentiable manifold
floor function here is a clue to the relevance of nilmanifolds to additive combinatorics: the so-called bracket polynomials, or generalised polynomials,
Nilmanifold
Hungarian mathematician (1941–2019)
the polynomial Freiman–Ruzsa conjecture, a central question of additive combinatorics, now also called Freiman's theorem. This was published by Imre Ruzsa
Katalin_Marton
3rd–2nd century BC Indian mathematician and poet
ISBN 978-0-19-562515-8. Shah, Jayant (2008). "A History of Piṅgala's Combinatorics" (PDF). Northeastern University, Boston. Archived from the original
Pingala
Theorem in combinatorics
In combinatorics, the Cameron–Erdős conjecture (now a theorem) is the statement that the number of sum-free sets contained in [ N ] = { 1 , … , N } {\displaystyle
Cameron–Erdős_conjecture
American mathematician and physics professor
devoted a section to Snevily's Conjecture in his well-known book Additive Combinatorics. Hunter collaborated the most with his long-term friend André Kézdy
Hunter_Snevily
Algebraic structure with addition, multiplication, and division
⋅ b = b ⋅ a. Additive and multiplicative identity: there exist distinct elements 0 and 1 in F such that a + 0 = a and a ⋅ 1 = a. Additive inverses: for
Field_(mathematics)
Property of some mathematical functions
in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. A subadditive function
Subadditivity
Mathematical problem
(2009). "Additive group theory and non-unique factorizations". In Geroldinger, Alfred; Ruzsa, Imre Z. (eds.). Combinatorial number theory and additive group
Zero-sum_problem
Theoretical object in mathematics
basis of an analogy between symmetries in projective geometry and the combinatorics of simplicial complexes. F1 has been connected to noncommutative geometry
Field_with_one_element
such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
U.S. lecture series in science and math
Izabella Laba, University of British Columbia, "Harmonic Analysis and Additive Combinatorics on Fractals" 2017 Talithia Williams, Harvey Mudd College, "Not So
AWM/MAA_Falconer_Lecture
Natural number
Diophantine geometry (Arakelov theory, Hodge–Arakelov theory) Arithmetic combinatorics (additive number theory) Arithmetic geometry (anabelian geometry, p-adic
1
matrix entries. Inverse Littlewood-Offord theorem: a result in additive combinatorics that addresses the structure of sets that exhibit concentration
List of Vietnamese inventions and discoveries
List_of_Vietnamese_inventions_and_discoveries
American mathematician
Folkman contributed important theorems in many areas of combinatorics. In geometric combinatorics, Folkman is known for his pioneering and posthumously-published
Jon_Folkman
Type of numeric sequence
ISBN 0-387-94655-1. Zbl 0859.11003. Tao, Terence; Vu, Van H. (2006). "Additive geometry". Additive Combinatorics. Cambridge University Press. ISBN 9780521853866.
Generalized arithmetic progression
Generalized_arithmetic_progression
Whole number
number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers
0
Recursive integer sequence
many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics: Volume 2 by combinatorialist
Catalan_number
Class of sequences of natural numbers
bibliography of work related to Sidon sequences". Electronic Journal of Combinatorics. 11 DS11: Jul 26: 39. doi:10.37236/32.. Guy, Richard K. (2004). "C9:
Sidon_sequence
Branch of elementary mathematics
Algorithmic Problems". In Tabachnikov, Serge (ed.). Kvant Selecta: Combinatorics, I: Combinatorics, I. American Mathematical Soc. ISBN 978-0-8218-2171-8. Vaccaro
Arithmetic
Design pattern in functional programming to build generic types
monad can be considered additive, with Nothing as mzero and a variation on the OR operator as mplus. List is also an additive monad, with the empty list
Monad (functional programming)
Monad_(functional_programming)
Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
of the characterization. The infinite cyclic group is isomorphic to the additive subgroup Z of the integers. There is one subgroup dZ for each integer d
Subgroups_of_cyclic_groups
Hungarian-American mathematician
contributions on additive number theory and ergodic theory" The Abel Prize citation also credited Szemerédi with bringing combinatorics to the centre-stage
Endre_Szemerédi
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