## Coloring problems

My student Tommy Chartier completed his M.S. in December. Here is a copy of the nice slides he used in his defense.

In his thesis, Coloring problems, Tommy studies some graph coloring problems. I wanted to emphasize the variety of problems that the title encompasses, so the examples he presents are all very different from one another.

1. First, Tommy talks briefly about the coloring number of the plane (the notoriously difficult Hadwiger-Nelson problem):

What is the smallest number of colors needed to color the plane, if no two points at distance 1 are to have the same color?

An excellent reference for this problem is the highly entertaining “The mathematical coloring book“, by Alexander Soifer.

There is a well-known finite graph, using 7 vertices, showing that at least 4 colors are needed. Tommy gives an argument showing that, on the other hand, any distance-1 graph with at most 6 vertices (in the plane) is 3-colorable. One can find this result in the literature; the reference is usually

• “Solution to problem 10” by L. Moser and W. Moser. Canadian Mathematics Bulletin, 4 (1961), 187-189.

Unfortunately, although the articles of the Bulletin are available online, problems and solutions are not, so we reproved it. It would be interesting to see how far one can generalize this result. I know of only one paper that has attempted this,

• “All unit-distance graphs of order 6197 are 6-colorable” by Dan Pritkin. Journal of Combinatorial Theory Series B, 73 (2) (1998), 159-163.

Besides the result in the title, Pritkin shows that any such graph of order at most 12 is 4-colorable. One should be able to improve this result with some work.

2. Tommy also presents some results we obtained during a reading course he took with me as an undergrad, on regressive Ramsey numbers, based on my paper:

in particular, Tommy presents his example showing that what I call $g(4,4)$ in the paper is precisely 85.

3. The main work on the thesis is on a problem that I find fascinating. The question (still open) originated on MathOverflow, and can be found here.

Is it true that for every $n$, we can color ${\mathbb Z}^+$ with $n$ colors, in such a way that, for every $a$, the numbers $a,2a,\dots,na$ all have different colors?

Given $n$, let $K_n$, the $n$-core, denote the set of positive integers whose prime factorization only involves primes less than or equal to $n$. As a first observation, the question above has a positive answer for $n$ iff it has a positive answer when ${\mathbb Z}^+$ is replaced with $K_n$. In fact, if there is a coloring of $K_n$ as in the question (a satisfactory $n$-coloring), then there are continuum many such colorings of ${\mathbb Z}^+$ using $n$ colors.

Tommy looks for sufficient conditions ensuring the existence of  satisfactory $n$-colorings for different $n$. The first condition (identified by Victor Protsak) is number theoretic:

Theorem 1. Suppose that there is a number $k$ such that $p=nk+1$ is prime, and the numbers $1^k,2^k,\dots,n^k$ are all different modulo $p$. Then the map $i\mapsto (i^k\mod p)$ is a satisfactory $n$-coloring.

If $n+1$ or $2n+1$ is prime, then the other condition is automatically satisfied. For other values of $k$, this is no longer the case. In fact, it is shown in Tommy’s thesis that $k$ can never be 3, for example, and that for any $k$ there are only finitely many possible $n$, see here.

There is a more general approach, suggested for example by Ewan Delanoy.

Definition. A partial ${\mathbb Z}_n$-homomorphism is a bijection $\varphi:\{1,\dots,n\}\to{\mathbb Z}_n$ such that whenever $ij\le n$, then $\varphi(ij)=\varphi(i)+\varphi(j)$.

Theorem 2. If there is a partial ${\mathbb Z}_n$-homomorphism, then there is a satisfactory $n$-coloring.

This is more general: If $p,k$ are as in Theorem 1, then the $k$-th powers modulo $p$ form a multiplicative group isomorphic to ${\mathbb Z}_n$, and a partial ${\mathbb Z}_n$-homomorphism can be easily built. On the other hand, Tommy exhibits examples of partial ${\mathbb Z}_n$-homomorphisms that do not come from any such prime $p$.

Even more generally, we can replace ${\mathbb Z}_n$ with any abelian group $(G,\cdot)$ of size $n$: If there is a bijection $\varphi:\{1,\dots,n\}\to G$ such that $\varphi(ij)=\varphi(i)\varphi(j)$ whenever $ij\le n$, then there is a satisfactory $n$-coloring. We call these induced colorings multiplicative.

The requirement that $G$ is abelian is a technical necessity to ensure a satisfactory coloring can be obtained from $\varphi$. It is interesting to ask whether there is such a bijection, even without this requirement. In

• “Groups formed by redefi ning multiplication”, by K. Chandler. Canadian Mathematics Bulletin, 31 (4) (1988), 419-423,

it is shown that $G$ must be abelian if $n$ is odd, but whether this is always the case seems to be open.

Moreover,

• “What is special about 195? Groups, $n$th power maps and a problem of Graham”, by R. Forcade and A. Pollington. Proceddings of the First Conference of the Canadian Number Theory Association, Ban , 1988, R.A. Mollin, ed., Walter de Gruyter, Berlin, 1990,

it is shown that $n=195$ is least such that there is no such bijection, for any $G$. Forcade provided us with the list of all $n\le 500$ for which there is no such bijection. We call such $n$ groupless. It is currently open whether any of them admit a satisfactory coloring.

Such colorings would by necessity be non-multiplicative and currently we do not have a framework to explain where they would come from, see here.

I have found examples of non-multiplicative colorings (for $n=6$), so we cannot currently rule out that there are satisfactory colorings for every $n$.  On the other hand, even though my examples are non-multiplicative, one can define multiplicative colorings from them. The situation for $n=195$ is very much open.

Many interesting questions remain. For example, it is possible that, for $n$ large enough, there are $p,k$ as in Theorem 1 iff either $n+1$ or $2n+1$ are prime. On the other hand, by a result of Kummer and Mills, see

• “Characters with preassigned values”, by W.H. Mills. Canadian Journal of Mathematics, 15 (1963), 169-171,

we know that for $n=34$ there must be such a $p$, but extensive computer search has not found it. As an indication of the difficulties of the search, the smallest $p$ for $n=32$ is $p=5209690063553$, and we expect any $p$ for $34$ to be orders of magnitude larger.

It is worth mentioning that the difficulties found with the problem were to be expected. As shown in the thesis, the existence of a satisfactory $n$-coloring trivially implies the case $m=n+1$ of the following result:

Theorem. (Balasubramanian-Soundararajan, 1996) Let $m\ge 1$, and let $0 be integers. Then $a_i/\gcd(a_i,a_j)\ge m$ for some $i,j$.

The Balasubramanian-Soundararajan theorem solved a conjecture of Graham from 1970. The proof appears in

• “On a conjecture of R. L. Graham”, by R. Balasubramanian and K. Soundararajan. Acta Arithmetica, 75 (1) (1996), 1-38.

The work of Forcade-Pollington mentioned above was carried out in connection with this conjecture.

It would be nice to revisit the problem in the future. I think there is still much to say about it.