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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).
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]
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]
The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3.; The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set.
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 ...
Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the ...
Add it up, and it's the worst two-game stretch in Los Angeles Lakers history. The Lakers lost to the Heat 134-93 on Wednesday, that loss coming two days after a 109-80 loss to the Timberwolves.
The American Mathematical Society provides a free online tool to determine the collaboration distance between two mathematical authors listed in the Mathematical Reviews catalogue. [11] Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 509 direct collaborators; [7] these are the people with Erdős number 1.