Open-problem statements, with their sources attached.

Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?

For all $n\ge 2k$ the least prime factor of $\binom{n}{k}$ is $\le\max(n/k,k)$, with only finitely many exceptions.

Szabo asks whether the maximal $t$ is given by

Ecklund, Erdős, and Selfridge [EES74] conjectured $g(k)\leq \exp((1+o(1))k)$.

Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 5$?

Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$.

Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$.

Does this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers?

Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$.

More generally, for which pairs $x$ and $A$ does this process terminate?

Is every odd $n > 1$ the sum of a squarefree number and a power of 2?

Graham has shown that $\frac{m}{n}$ is the sum of distinct unit fractions with denominators $\equiv a\pmod{d}$ if and only if $\left(\frac{n}{(n,a,d)},\frac{d}{(a,d)}\right)=1.$ Does the greedy algorithm always terminate in such cases?

Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4.

Graham has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$. Does the greedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it fails to terminate almost always.

Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?

Let $k(N)$ denote the smallest $k$ such that there exists $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$

Is it true that $\lim_{N \to \infty} k(N) - (e - 1)N = \infty$?

1. There is NO good sequence with polynomial growth.

There are no examples known of the weakened coprime version if we insist that $1\not\in P\cup Q$.

1. There is a good sequence with sub-exponential growth.

Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with $\sum_{n \in A} 1/n > K$ there exists some $S \subseteq A$ such that $1 - \exp(-(c*K)) < \sum_{n \in S} 1/n \le 1$?

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.