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Let $n$ be a positive integer, and let $n = \ell_1 + \dots + \ell_k$ be a partition of $n$. Then there exists a convex polygon with side lengths $\ell_1, \dots, \ell_k$ if and only if all of the $\ell …

Yes, there is a common name for such graphs -- they are called graphs with polynomial growth. See e.g. W. Imrich, N. Seifter: A survey on graphs with polynomial growth, Discr. Math. 95 (1991), 101-11 …

Bounds on Rado numbers for your equation can be found in: Brian Hopkins, Daniel Schaal: On Rado numbers for $\sum_{i=1}^{m-1} a_i x_i = x_m$, Adv. in Appl. Math. 35(2005), no. 4, 433-441.

Is it true that given a finitely presented group $G$, either all primes or only finitely many of them occur as orders of elements of $G$?

For small enough $n$, an efficient way to perform this enumeration is described in the solution to a GAP exercise I posed a few years ago. It basically amounts to setting up a suitable matrix, raising …

On page 218 of John D. Dixon, Brian Mortimer: Permutation Groups, Springer GTM 163, 1996 it is stated: It is a consequence of the classification of finite simple groups that a finite permutat …

Today I noticed that the last relator in the 27-relator presentation of a group with unsolvable word problem given in Donald J. Collins: A simple presentation of a group with unsolvable word problem. …

There are indeed other such polygons. -- For example there is one for $n = 11$, as follows (the origin is in the lower left corner): Also there is one for $n = 15$: Further there are $21$ such p …

Me funksionin GAP MaxOnes := n -> Maximum(List(Filtered(AsList(GF(2)^[n,n]), M->not ForAny(Tuples([1..n-1],2), s->ForAny(Cartes …

It seems that a nice example is the set $A$ of positive integers which have an even number of 1's in their binary expansion, although I don't see a reasonable lower bound on $R_k(A)$ for now. A quick …

Let $f$ denote the function described in the question. The assertion that every trajectory of $f$ except for the one starting at 0 ends in the cycle -1, 1, -1 is equivalent to the Collatz conjecture s …

Is there a bound $B$ such that every 2-generator subgroup $G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$ whose generators do not satisfy a relation of length $\leq B$ is free? If it exists, su …

If you are also interested in problems of that type where $n = \infty$: Given a mapping $f: \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to themselves, it is often a notoriously hard pr …

For the sake of simplicity, consider only the case $d=2$. In this case, two pairs $(a,b), (a,c) \in {\rm S}_n^2$ lie in the same orbit if and only if there is a permutation $\pi$ in the centralizer of …

Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ …