Open-problem statements, with their sources attached.

In particular, the Dedekind number for n = 10 is currently unknown.

Is there some set $A\subset\{1, ..., N\}$ of size $\geq (1 - o(1))N$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$?

Babai–Seress Conjecture (Conjecture 1.5)**: There exists an absolute constant $C$ such that the diameter of the alternating group $A_n$ satisfies $\operatorname{diam}(A_n) \leq n^C.$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.5

Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $g(n) > n^{1-\epsilon}$?

Babai–Seress Conjecture (Conjecture 1.7)**: There exists an absolute constant $C$ such that every finite simple non-abelian group $G$ satisfies $\operatorname{diam}(G) \leq (\log |G|)^C.$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.7

Are there infinitely many $n$ such that, for all $k\geq 1$

Every tree admits a graceful labeling.

A graceful labeling of a tree $T$ with $m$ edges is an injective map $f : V \to \{0, \dots, m\}$ such that the multiset of absolute differences $|f(u) - f(v)|$ over edges $\{u,v\}$ of $T$ equals $\{1, \dots, m\}$.

Is it true that, for any $a \in \mathbb{Z}$, there are infinitely many $n$ such that $\phi(n) | n + a$?

The Komlós conjecture*

There exists a universal constant $K > 0$ such that for all $n, m \in \mathbb{N}$ and all vectors $v_1, \dots, v_n \in \mathbb{R}^m$ with $\|v_i\|_2 \le 1$ (encoded here as $\sum_j v_{ij}^2 \le 1$), there exist signs $\varepsilon_i \in \{-1, +1\}$ such that $\left\|\sum_i \varepsilon_i v_i\right\|_\infty \le K$, i.e. $\left|\sum_i \varepsilon_i v_{ij}\right| \le K$ for every coordinate $j$.

When $n > 1$, Lehmer conjectured that $\phi(n) | n - 1$ if and only if $n$ is prime.

Does there exist a $3 \times 3$ semi-magic square whose entries are all distinct positive integer cubes? A square is semi-magic if all rows and columns sum to the same total.

More precisely, we seek a $3 \times 3$ matrix with entries $a_{ij}$ such that each $a_{ij} = n_{ij}^3$ for some positive integer $n_{ij}$, all nine cubes are distinct, and all row sums and column sums are equal. Reference:* Semi-Magic Square of Cubes

Erdős Problem 829 (open).* Let $A \subseteq \mathbb{N}$ be the set of perfect cubes. Is it true that $(1_A \ast 1_A)(n) \ll (\log n)^{O(1)}$? That is, does there exist a natural number $C$ such that the number of representations of $n$ as a sum of two cubes is $O((\log n)^C)$ as $n \to \infty$?

The pebbling number conjecture: the pebbling number of a Cartesian product of connected graphs is at most equal to the product of the pebbling numbers of the factors. See Asplund, Hurlbert, and Kenter.

Erdos Problem 830, Part 1* We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. Are there infinitely many amicable pairs?

The open problem: determine the Ramsey number $R(5,5)$.

It is known that $43 \le R(5,5) \le 46$.

Erdos Problem 830, Part 2* We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then is it true that $A(x) > x^{1-o(1)}?$

Sidorenko's conjecture (1993).*

For every finite bipartite simple graph $H$ and every finite simple graph $G$: $t(H, G) \ge t(K_2, G)^{e(H)}$, where $K_2$ denotes the single-edge graph on 2 vertices (i.e. completeGraph (Fin 2)).

Is it true that, for every integer $t\geq1$, there is some integer $a$ such that ${n \choose k} = a$ with $1\leq k \le \frac{n}{2}$ has exactly $t$ solutions?

For dimension $9$, the length of the longest snake in the box is not known. This is currently the smallest dimension where this question is open.

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?