Open-problem statements, with their sources attached.

What is the smallest square that can contain 21 unit circles?

What is the smallest circle that can contain 15 unit circles?

Reference: Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.

Moser's Worm Problem* What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?

Convex Moser's Worm Problem* What is the minimal area (or greatest lower bound on the area) of a convex shape that can cover every unit-length curve?

The Beck–Fiala conjecture*

There exists a universal constant $C > 0$ such that every set system $S_1, \dots, S_m \subseteq [n]$ of degree at most $t$ admits a colouring $\chi \colon [n] \to \{-1, +1\}$ with $\left|\sum_{j \in S_i} \chi(j)\right| \le C \sqrt{t}$ for every $i$.

Does there exist an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors? Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle. The first condition is equivalent to being locally linear.

No closed-form expression that allows efficient computation of Dedekind numbers is currently known.

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

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

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

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 $\{1, \dots, m\}$.

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 $j$.

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

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.

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

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

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)).

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.

Wichmann's conjecture on optimal rulers.* Every optimal ruler with more than $13$ segments is a Wichmann ruler $W(r, s)$ (up to reflection, i.e. reversing the segment list). Posed by Wichmann [Wi63]; the finitely many known exceptions all have at most $13$ segments (lengths $1, 13, 17, 23, 58$), and no further exceptions are known up to length $213$.

Construct an $S(t, k, n)$-Steiner system with $n > k > t > 5$, $t < 10$, and $n < 200$.

No example of a Steiner system with $t > 5$ is known, despite a 2014 existence theorem by Keevash showing that such systems must exist for sufficiently large $n$. Reference:* Large Steiner Systems

For every finite union-closed family of sets, other than the family containing only the empty set, there exists an element that belongs to at least half of the sets in the family.