In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. (Here R(r, s) signifies an integer that depends on both r and s.)
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s.
Sir William Timothy Gowers, is a British mathematician. He is Professeur titulaire of the Combinatorics chair at the Collège de France, and director of research at the University of Cambridge and Fellow of Trinity College, Cambridge. In 1998, he received the Fields Medal for research connecting the fields of functional analysis and combinatorics.
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.
In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.
Ben Joseph Green FRS is a British mathematician, specialising in combinatorics and number theory. He is the Waynflete Professor of Pure Mathematics at the University of Oxford.
A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-st and the n-th prime numbers, i.e.
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949. Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to be the ratio area/sys2. The systolic ratio SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry.
In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture about the long-term behavior of runners on a circular track. It states that runners on a track of unit length, with constant speeds all distinct from one another, will each be lonely at some time—at least units away from all others.
The Erdős–Szemerédi theorem in arithmetic combinatorics states that for every finite set of integers, at least one of , the set of pairwise sums or , the set of pairwise products form a significantly larger set. More precisely, the Erdős–Szemerédi theorem states that there exist positive constants c and such that for any non-empty set
Nets Hawk Katz is the IBM Professor of Mathematics at the California Institute of Technology. He was a professor of Mathematics at Indiana University Bloomington until March 2013.
Tom Sanders is an English mathematician, working on problems in additive combinatorics at the interface of harmonic analysis and analytic number theory.
Wojciech Samotij is a Polish mathematician who works in combinatorics, additive number theory, Ramsey theory and graph theory.
In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axis-aligned rays with a shared apex.
József Balogh is a Hungarian-American mathematician, specializing in graph theory and combinatorics.
Thomas F. Bloom is a mathematician, who is a Royal Society University Research Fellow at the University of Oxford. He works in arithmetic combinatorics and analytic number theory.
Hunter Snevily (1956–2013) was an American mathematician with expertise and contributions in Set theory, Graph theory, Discrete geometry, and Ramsey theory on the integers.
