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One can interpret the positions of the numbers in a sequence as x-coordinates of points in the Euclidean plane, and the numbers themselves as y-coordinates; conversely, for any point set in the plane, the y-coordinates of the points, ordered by their x-coordinates, forms a sequence of numbers (unless two of the points have equal x-coordinates).
To find a solution for , just divide all of the unit fractions in the solution for by : = + + = + +. If 4 n {\displaystyle {\tfrac {4}{n}}} were a counterexample to the conjecture, for a composite number n {\displaystyle n} , every prime factor p {\displaystyle p} of n {\displaystyle n} would also provide a counterexample 4 p {\displaystyle ...
Barker's notation refers to the ERD notation developed by Richard Barker, Ian Palmer, Harry Ellis et al. whilst working at the British consulting firm CACI around 1981. The notation was adopted by Barker when he joined Oracle and is effectively defined in his book Entity Relationship Modelling as part of the CASE Method series of books.
The idea of the Erdős number was originally created by the mathematician's friends as a tribute to his enormous output. Later it gained prominence as a tool to study how mathematicians cooperate to find answers to unsolved problems. Several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy. [7]
More precisely, Mycielski (1961) showed that the theorem is a consequence of the Boolean prime ideal theorem, a property that is implied by the axiom of choice but weaker than the full axiom of choice, and Läuchli (1971) showed that the De Bruijn–Erdős theorem and the Boolean prime ideal theorem are equivalent in axiomatic power. [15]
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If H is an arbitrary graph whose chromatic number is r > 2, then H is contained in K r (t) whenever t is at least as large as the largest color class in an r-coloring of H, but it is not contained in the Turán graph T(n,r − 1), as this graph and therefore each of its subgraphs can be colored with r − 1 colors.
Paul Erdős, Chao Ko, and Richard Rado proved this theorem in 1938 after working together on it in England. Rado had moved from Berlin to the University of Cambridge and Erdős from Hungary to the University of Manchester, both escaping the influence of Nazi Germany; Ko was a student of Louis J. Mordell at Manchester. [6]