Directed Graph Ramsey Numbers
It is also possible to define Ramsey numbers for directed graphs. (These were introduced by P. Erdős & L. Moser.) Let R(n) be the smallest number Q such that any complete graph with singly directed arcs (also called a "tournament") and with ≥ Q nodes contains an acyclic (also called "transitive") n-node subtournament.
This is the directed-graph analogue of what (above) has been called R(n,n;2), the smallest number Z such that any 2-colouring of the edges of a complete undirected graph with ≥ Z nodes, contains a monochromatic complete graph on n nodes. (The directed analogue of the two possible arc colours is the two directions of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way," i.e. "acyclic.")
Indeed many find the directed graph problem to actually be more elegant than the unidirected one. We have R(0)=0, R(1)=1, R(2)=2, R(3)=4, R(4)=8, R(5)=14, R(6)=28, 32≤R(7)≤55, and R(8) is again a problem you do not want powerful aliens to pose.
Read more about this topic: Ramsey's Theorem
Famous quotes containing the words directed, graph and/or numbers:
“Religion, or the duty which we owe our Creator, and the manner of discharging it, can be directed only by reason and conviction, not by force and violence; and therefore all men are equally entitled to the free exercise of religion, according to the dictates of conscience.”
—James Madison (17511836)
“When producers want to know what the public wants, they graph it as curves. When they want to tell the public what to get, they say it in curves.”
—Marshall McLuhan (19111980)
“I had a feeling that out there, there were very poor people who didnt have enough to eat. But they wore wonderfully colored rags and did musical numbers up and down the streets together.”
—Jill Robinson (b. 1936)