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Revision of Existence proofs from Thu, 18/12/2008 - 18:29

Quick description

Many mathematical problems require one to prove that a mathematical object with certain properties exists. There are many different techniques for proving existence statements. This page briefly describes some of these and gives links to more detailed discussions.

General discussion

A large number of mathematical problems require one to prove that a mathematical object with certain properties exists. More ... ( Sometimes this is a very direct requirement of the problem: for example, a well-known problem (now solved) is whether there exists a polynomial in several real variables that takes only positive values but that cannot be written as a sum of squares of other polynomials. (The answer is yes.) But even more common are problems that are logically of the form "For all X there exists Y such that ***." Such a problem can be regarded as an existence problem with X treated as a parameter. For instance, a well-known question in graph theory asks whether for every pair of positive integers r and m there exists a graph with chromatic number greater than r, which means that there is no way of colouring the vertices with r colours without there being an edge that joins two vertices of the same colour, and girth at least m, which means that there is no cycle of length less than m. Here r and m can be thought of as fixed (but unspecified), and the problem is then to determine whether there exists a graph with a certain property that is defined in terms of r and m. (The answer is again yes, whatever the values of r and m.) There are many techniques for proving existence statements, and they can be strikingly different from one another. In this article we give links to articles about some of these techniques. For a quick indication of what an article is about, click on "Brief summary"). We continue with a general discussion about how to decide which of these techniques is most likely to help with the existence statement you are trying to prove. At the end of the article is a list of links to more specific articles about proving the existence of particular kinds of mathematical object. )

Technique 1: off-the-shelf examples Brief summary ( Suppose that you are asked to prove that a mathematical object X of a given kind (such as a prime number, or a group, or a function from the real numbers to the real numbers) exists with certain properties. The most obvious method is probably to look at familiar examples of that kind of mathematical object and see if one of them has the property in question. For example, does there exist a function from the reals to the reals that is non-constant, bounded, and infinitely differentiable? Most people would answer that question by having a thought such as, "Polynomials don't work, so what's the next simplest sort of function? I could try e^x but that doesn't work either because it's unbounded. Ah, I could use \sin x or \cos x." It is unlikely that you will be able to solve an unsolved existence problem by simply searching around for a known example that does the job for you, unless the difficulty lies in showing that the known example has the property you want. However, not all existence problems arise directly as unsolved problems. For instance, you might be thinking about another problem and realize that it would be helpful if you could prove a certain lemma of the form "For every X there exists Y such that ..." Then you would be faced with a "mini-problem" of a kind that comes up a lot in research. You would not necessarily know in advance whether it was easy or hard, so it would be worth searching for an off-the-shelf construction of Y rather than starting from scratch. It also often happens that a non-trivial counterexample to one conjecture can be modified to disprove several others: it is worth getting to know the important counterexamples in your area in case you can use them for your own purposes in another context. )

Technique 2: building a bespoke example Brief summary ( Another very obvious idea for proving existence statements is to design examples specifically to have the properties that are being asked for. For instance, Euclid's proof that there are infinitely many primes does this: he assumes that there are finitely many primes p_1,p_2,\dots,p_k and then designs the number n=p_1p_2\dots p_k+1 not to be a multiple of any of those primes. )

Technique 3: making one mathematical structure out of another Brief summary ( There are many ways of building one kind of mathematical object out of another. For example, given a graph G with n vertices one can define an n\times n symmetric matrix A by taking A_{xy} to be 1 if xy is an edge of G and 0 otherwise. Or given a connected manifold one can define its fundamental group. Or given a group one can define its group algebra. The list of ways in which one object can be used in the construction of another is very long. )

Technique 4: universal examples with given properties Brief summary ( Suppose you are asked to prove that a mathematical object X of a certain type can be found that satisfies some property P. It quite often happens that there is a partial order on the given kinds of objects such that if X is less than Y and X has property P, then so does Y. If that is the case, then one can look for objects that are maximal in this partial order. This observation leads to the notion of a universal construction. )

Technique 5: indirect proofs of existence

Technique 6: the probabilistic method Brief summary ( If you are trying to prove that an object X of type \tau exists with certain properties, and if those properties seem to force X to be "spread about" and "unstructured", then a good method may well be to define a simple probability distribution on all objects of type \tau and prove that there is a non-zero probability that a randomly chosen X has the desired properties. )

Technique 7: cardinality, measure and category

How do I decide which technique to use?

It is difficult to give a definitive answer to this question, but there are certainly features to look out for when one is trying to assess which style of proof is most likely to be suitable for a given existence problem. Looking for an example amongst a list of well-known constructions is usually best if you have only just come across the problem and have no evidence to suggest how hard it is. One can think of this as a preliminary test: if you don't come across an example, then the act of establishing that well-known examples don't work may at least provide a few clues about how to go about the proof.

If off-the-shelf examples don't solve your problem, what should you do next? Here are some questions that can help with the decision. Suppose that your problem is of the form "Does there exist an X such that P(X)?" Here X is some type of mathematical object, like a field or an analytic function from \mathbb{C} to \mathbb{C}, and P is some property that X may or may not have. Then it can make a big difference if P can be split up into several different properties. For example, returning to Euclid's proof that there are infinitely many primes, he assumes that the only primes are p_1,\dots,p_k, and is faced with the question, "Does there exist n such that n is not a multiple of any prime?" If it were not fairly easy to come up with an example of such an n (under the hypothesis that will thereby be contradicted), then it would be possible to exploit the fact that this property splits up into the k properties " n is not a multiple of p_i" and prove the result as one proves the Chinese remainder theorem. The idea would be to start by finding a number n_1>1 that wasn't a multiple of p_1 and remember that adding any multiple of p_1 to it would give another non-multiple of p_1. So if n_1 was a multiple of p_2, one could add p_1 to it and the result would no longer be a multiple of p_2 since p_2 is not a factor of p_1. To this new number n_2 one could add any multiple of p_1p_2 and it would still not be a multiple of either p_1 or p_2. So if n_2 was a multiple of p_3 one could add p_1p_2 to it and the result would no longer be a multiple of p_3, since p_3 is not a factor of p_1p_2. (This is a fairly deep fact, however.) And so on all the way up to p_k. Of course, the usual proof of just exhibiting the number p_1p_2\dots p_k+1 is simpler for this problem, but the fact that we can split the property up as a conjunction of several simpler properties does at least show that there is a way of solving the problem that doesn't need one to have that idea and just deals with the properties one by one. To summarize, you should ask yourself: does the property I am looking for split up as the conjunction of several simple properties?

If the answer is yes, then there are other very important questions you should ask. A very obvious one is: Do I have finitely many constraints or infinitely many? And a less obvious but perhaps more important one: Are the simple properties "loose constraints" on X or "tight constraints" on X? And a closely related question: Does X have many degrees of freedom or few?

Roughly speaking, a loose constraint is one that does not cut down by too much the space of all possibilities for X. For example, if you are trying to find a sequence of integers that satisfies properties P_1,P_2,P_3\dots, and property P_k requires that every term of the sequence beyond a certain point N_k is at least as big as m_k, then that is probably a loose constraint, since there is still a vast amount of flexibility in how you choose your sequence. (I say "probably" because it might conceivably be that in the presence of the other properties that the sequence is required to have, this property does hugely cut down the possibilities. But it is usually quite easy to recognise a loose constraint when you have one.) Having loose constraints is helpful if the object X that you want to exist can be built up in stages. In the case of a sequence, it is obvious that this can be done: one builds up the sequence an element at a time. But it also applies to less clearly composite objects such as real numbers, which can be defined as limits of sequences. And as the discussion of Euclid's proof showed, even a positive integer can on occasion be built up in stages.

Sometimes one can prove that there is an object X with many properties P_i by proving that for every i the set A_i of objects that do not have property P_i is small. If the sets A_i are small enough in a suitable sense (that varies from problem to problem) then one may be able to show that their union cannot include all objects X. If objects of type X have finitely many degrees of freedom and one wants to find X with finitely many properties P_1,\dots,P_N, then one may be able to assign sizes to each set of objects without property P_i and show that these sizes add up to less than the size of the set of all objects of type X. This is how the probabilistic method works. If there is a countably infinite collection of properties, then one often tries to find a notion of smallness with the following two features: a countable union of small sets is small; the set of all objects of type X is not small. Then if the set A_i of all objects of type X without property P_i is small for each i, it follows that there is some X that has all the properties P_i. Some common notions of smallness are |being countable, having measure zero, and being of the first category.

Parent article

What kind of problem am I trying to solve?

Offspring of this article

This article is a very general one about proving existence statements. However, there are other articles devoted to more specific topics, such as proving the existence of groups with certain properties, or graphs, or sets of integers, or normed spaces, or functions, etc. There is considerable overlap between those articles and this general one, but if you know that it is a group that you want to construct, then it may be more efficient to look at the article about existence of groups rather than this general article about existence.

Proving the existence of groups with certain properties

Proving the existence of sets of integers with certain properties

Proving the existence of finite subsets of groups with certain properties

Proving the existence of graphs with certain properties

Proving the existence of functions with certain properties