For you to be certain that you will have four points, every pair of which is connected by a red line, or four points, every pair of which is connected by a green line, you will need 18 points, and for you to be certain that there will be five points with this property, you will need -- it's not known exactly - between 43 and 55. With enough points, you will inevitably find unicolored islands of order as big as you want, no matter how you color the lines.
A whole mathematical subdiscipline has grown up devoted to proving theorems of this same general form: How big does a set have to be so that there will always be some subset of a given size that it will constitute an island of order of some sort?
Ramsey-type theorems may even be part of the explanation (along, of course, with Diaconis' dictum) for some of the equidistant letter sequences that constitute the bible codes. Any sufficiently long sequence of symbols, especially one written in the restricted vocabulary of ancient Hebrew, is going to contain subsequences that appear meaningful.
Finally, of more direct relevance to evolution and the origin of living complexity is the work of Stuart Kauffman. In his book, "At Home in the Universe," he discusses what he has termed the aforementioned notion of "order for free."
Motivated by the idea of hundreds of genes in a genome turning on and off other genes and the order and pattern that nevertheless exist, Kauffman asks us to consider a large collection of 10,000 light bulbs, each bulb having inputs from two other bulbs in the collection.
Assume that you connect these bulbs at random, that a clock ticks off one-second intervals, and that at each tick each bulb either goes on or off according to some arbitrarily selected rule. For some bulbs, the rule might be to go off at any instant unless both inputs are on the previous instant. For others it might be to go on at any instant if at least one of the inputs is off the previous instant. Given the random connections and random assignment of rules, it would be natural to expect the collection of bulbs to flicker chaotically with no apparent pattern.
What happens, however, is that very soon one observes order for free, more or less stable cycles of light configurations, different ones for different initial conditions. Kauffman proposes that some phenomenon of this sort supplements or accentuates the effects of natural selection.
Although there is certainly no need for yet another argument against the seemingly ineradicable silliness of "creation science," these light bulb experiments and the unexpected order that occurs so naturally in them do seem to provide one.
In any case, order for free and apparent complexity greater than we might naively expect are no basis for believing in God as traditionally defined. Of course, we can always redefine God to be an inevitable island of order or some sort of emergent mathematical entity. If we do that, the above considerations can be taken as indicating that such a pattern will necessarily exist, but that's hardly what people mean by God.
John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as of the just-released "Irreligion: A Mathematician Explains Why The Arguments for God Just Don't Add Up " His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.