combinatorics

Latin squares are arrangements of symbols in a grid in which every symbol appears exactly once in each row and column. These symbol arrangements, which were first studied more than three centuries ago, are now widely used to optimize experimental designs and develop secure cryptographic systems, puzzles or other complex combinatorial structures. In 1782, the […]

This is the first article in a series about a discovery I made while researching Tserouf in the writings of the Kabbalist Abraham Aboulafia (1240 – after 1291). Tserouf is the Kabbalistic art of permuting the letters — and, in plain mathematical terms, it is the enumeration of all the permutations of an n-letter word. The context In Or ha-Sekhel (“The Light of the Intellect”), Aboulafia prescribe…

Suppose that one has a set of points in the plane, which we will think of as the complex plane . Let denote the number of unit distances determined by these points, i.e., pairs of points whose displacement obeys the equation (It makes little difference for the asymptotics, but we will count the pair separately […]

Ordnung muss sein! (Order is inevitable.) This saying , attributed to Theodore Motzkin, aptly summarizes Ramsey theory , which aims to solve problems of the form: “From what size on can a structure no longer avoid having a certain property ?"

The n queens problem is to place on an n × n chessboard n queens so that none attacks any other. This means there is only one queen on every horizontal, vertical, and diagonal line. When n is a prime number ≥ 5, it is sufficient to place the queens on a line that has slope 2, 3, 4, …, […] The post Queens on a prime order board first appeared on John D. Cook .

Fix integers and and set . Let denote the complete -partite -uniform hypergraph with parts of size . We prove that the Zarankiewicz number provided . Previously this was known only for due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of , for example, it gives where the exponent 11/4 is …