5 problems

For $k,N\in\mathbb{N}$, call a finite set $A\subseteq\{1,\dots,N\}$ $k$-weakly divisible if every $k+1$ elements of $A$ are not pairwise relatively prime. Let $M_k(N)$ be the large…

For $N\in\mathbb N$, let $E_N=\{n\in\{1,\ldots,N\}:2\mid n\text{ or }n\text{ is not squarefree}\}$. Is it true that, for every $N$, the maximum cardinality of a finite set…

Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some element $x$ such that whenever…

For a positive integer $N$, what is the maximum size of a set $A⊆{1,…,N}$ such that $N∈A$ and $\gcd(a,b)>1$ whenever $a,b∈A$ are distinct?

Problem of Lovász and myself, [32]. Let $f(n)$ be the smallest integer with the following property: There is a family $A_k$, $1 \leqq k \leqq f(n)$ satisfying $|A_k| = n$,…