In the picture-hanging puzzle we are to hang a picture so that the string loops around $n$ nails and the removal of any nail results in a fall of the picture. We show that the length of a sequence representing an element in the free group with $n$ generators that corresponds to a solution of the picture-hanging puzzle must be at least $n2^{sqrt{log_2 n}}$. In other words, this is a lower bound on the length of a sequence representing a non-trivial element in the free group with $n$ generators such that if we replace any of the generators by the identity the sequence becomes trivial.
In 1990, Alon and Kleitman proposed an argument for the sum-free subset problem: every set of n nonzero elements of a finite Abelian group contains a sum-free subset A of size |A|>frac{2}{7}n. In this note, we show that the argument confused two different randomness. It applies only to the finite Abelian group G = (Z/pZ)^s where p is a prime. For the general case, the problem remains open.
Let $S$ be a connected graph which contains an induced path of $n-1$ vertices, where $n$ is the order of $S.$ We consider a puzzle on $S$. A configuration of the puzzle is simply an $n$-dimensional column vector over ${0, 1}$ with coordinates of the vector indexed by the vertex set $S$. For each configuration $u$ with a coordinate $u_s=1$, there exists a move that sends $u$ to the new configuration which flips the entries of the coordinates adjacent to $s$ in $u.$ We completely determine if one configuration can move to another in a sequence of finite steps.
The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify some variants of the Frankl conjecture.
There is a bijection from Schroder paths to {4132, 4231}-avoiding permutations due to Bandlow, Egge, and Killpatrick that sends area to inversion number. Here we give a concise description of this bijection.
This exposition contains a short and streamlined proof of the recent result of Kwan, Letzter, Sudakov and Tran that every triangle-free graph with minimum degree $d$ contains an induced bipartite subgraph with average degree $Omega(ln d/lnln d)$.