Open-problem statements, with their sources attached.

BMO#8

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be two sequences such that $(a_1, b_1) = (10, 12)$ and

Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 1.

For every $n$ there exists some $d$ such that every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most $d$.

If $n \ge 2$, every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most 1.

If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.

For even m > 2, it is open whether the cube digraph on (ZMod m)³ has a Hamiltonian arc decomposition.

If x is a fusible number and y is its successor, then the interval [x + 1, y + 1) can be divided into intervals [ℓₙ, ℓₙ₊₁), such that the fusible numbers in [ℓₙ, ℓₙ₊₁) are obtained by fusing the n + 1st successor of x with a fusible number. This formalization differs from Conjecture 7.1 in the paper in four ways: (1) it is obtained from Conjecture 7.1 by plugging in n + 1 into n, which simplifies the expressions and removes the need to assume n ≥ 1; (2) the n + 1st successor s^(n+1)(x) is replaced by the explicit value x + (2 - 1 / 2 ^ n) * m; (3) instead of defining y to be the successor of x, we assert that there is no fusible number strictly between x and y; (4) instead of using ∃ z, IsFusible z ∧ q = s^(n+1)(x) ~ z we use the value of z determined by the equality, namely z = 2 * q - 1 - s^(n+1)(x), and it is easy to see z ∈ [x + 1 - m / 2 ^ n, x + 1) as required.

For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of $T$ via cyclic shifts of a single embedding.

The $2n+1$ copies are $f_0, f_1, \dots, f_{2n}$ where $f_i(v) = f_0(v) + i$ for all vertices $v$ , each copy is obtained by adding $i \pmod{2n+1}$ to every vertex of the base copy. This is strictly stronger than RingelConjecture.ringel_conjecture.

Conjecture 3.2 in [Wa2011]: Each Latin square of odd order has at least one transversal.

The smallest odd number for which this conjecture is not known is 11.

Conjecture 5.1 in [Wa2011]: Every latin square has a near-transversal

Conjecture 6.7 in [Wa2011]: There exist real constants $0 < c_1 < c_2 < 1$ such that

for all odd $n \geq 3$.

Conjecture 6.9 in [Wa2011]:

It is not even known if this limit exists. Note that $z_n = 0$ for even $n$ (see z_even), so the limit must be restricted to odd $n$; here we parametrise odd $n$ as $2k + 1$.

MOLS existence problem: determine exactly which orders n admit a complete set of n - 1 mutually orthogonal latin squares.

Equivalently, this asks for which orders affine planes of order n exist. Complete sets are known for prime-power orders; the smallest currently unresolved order is 12.

The smallest unresolved case of the MOLS existence problem: whether there are 11 mutually orthogonal latin squares of order 12.

The Latin Tableau Conjecture: If G is the simple graph of a Young diagram, then G is CDS-colorable.

For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

For $N = 6$ and $D = 4$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

For $N = 6$ and $D = 5$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?

For $N = 6$ and all $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?