In the study of mathematical analysis and calculus, a geometric series is a series in which the ratio of successive adjacent terms is constant. In other words, the sum of consecutive terms of a geometric sequence forms a geometric series. Thus, each term is the geometric mean of its two neighboring terms, similar to how the terms in an arithmetic series are the arithmetic means of their two neighboring terms.
Euler's constant is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
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.
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.
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.
The mathematical field of combinatorics was studied to varying degrees in numerous ancient societies. Its study in Europe dates to the work of Leonardo Fibonacci in the 13th century AD, which introduced Arabian and Indian ideas to the continent. It has continued to be studied in the modern era.
Nets Hawk Katz is the W.L. Moody Professor of Mathematics at Rice University. He was a professor of mathematics at Indiana University Bloomington until March 2013 and the IBM Professor of Mathematics at the California Institute of Technology until 2023. He is currently the W. L. Moody Professor of Mathematics at Rice University.
Tom Sanders is an English mathematician, working on problems in additive combinatorics at the interface of harmonic analysis and analytic number theory.
In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other. A topological graph is also called a drawing of a graph.
Wojciech Samotij is a Polish mathematician and a full professor at the School of Mathematical Sciences at the Tel Aviv University. He is known for his work 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 Manchester. 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.
Julian Sahasrabudhe is a Canadian mathematician who is an assistant professor of mathematics at the University of Cambridge, in their Department of Pure Mathematics and Mathematical Statistics. His research interests are in extremal and probabilistic combinatorics, Ramsey theory, random polynomials and matrices, and combinatorial number theory.
