58 problems

Let $L$ be a random Latin square of order $n$, and let $cmathbb{E}_a(n)$ denote the expected number of subsquares of order $a$ in $L$. For $2\leqslant a\leqslant n/2$, McKay and Wa…

Let $S_n$ be the set of permutations of $[n]$. A permutation is $k$-universal if it contains every permutation in $S_k$, and choose a permutation uniformly at random from $S_n$. Al…

For $K>0$, a $k\times n$ Latin rectangle is $K$-sparse if every row contains at most $K$ non-empty cells, every column contains at most $K$ non-empty cells, and every symbol occurs…

For a non-empty partial Latin square $P$, define its density by … Let $\mathbf{L}$ be a random Latin square of order $n$. The density threshold conjecture. As $n\to\infty$, … When…

A pseudorandom class of hypergraphs is a hereditary class whose sparse random model contains, with high probability, an induced subhypergraph on at least $(1-o_d(1))n$ vertices bel…

Bivariate normal limit conjecture. The centralized-scaled pair tends, in distribution, to the bivariate normal distribution with covariance

Let $r\geqslant 3$, let $\mathcal{P}$ be a set of three pairwise non-harmonious $r$-patterns with at least one collectable pattern, and let $z_{\mathcal{P}}(\mathbb{RM}^{(r)}_n)$ d…

Let $n$ be even, let $K_n$ be the complete graph on $n$ people with independent uniformly random preference lists, and let $X$ be the number of stable matchings. Mertens' conjectur…

Let $n$ be even, let $K_n$ be the complete graph on $n$ people, and let each person choose an independent uniformly random strict preference ordering of the other $n-1$ people. Let…

Let $\mathcal{P}^{(4)}_{\mathrm{Dyck}}$ be the set of Dyck $4$-patterns, and let … For the random ordered $4$-uniform hyper-matching $\mathbb{RM}^{(4)}_{n}$, let…

Let $r$ be a positive integer and let $\mathcal{P}$ be a set of three $r$-patterns that are pairwise non-harmonious, with at least one pattern collectable. For a random ordered $r$…

General conjecture. For every integer $d d45a 2$, there is a positive real number $C=C(d)$ such that, as the prime power $q$ tends to infinity, with high probability,

Let $D$ be an even natural number and let $G= obreakfrac{}{}mathcal{L}(K{frac{D}{2}+1,frac{D}{2}+1})$ be the line graph of a complete bipartite graph. Thus $G$ has…

Pak–Panova's conjecture. With high probability,

Pak–Panova's conjecture. With high probability,

Let $k\geq 3$ be a positive integer and let $0<\delta,\varepsilon<1$ be positive real numbers. There is an $n_0$ such that, for every $n\geq n_0$, if $H$ is an $n$-vertex $k$-graph…

Let $L$ be a Latin square of order $n$, and let $\ell(L)$ denote the length of its longest row cycle. Longest row-cycle conjecture. As $n\to\infty$, asymptotically almost all Latin…

Let $\mathcal{U}$ be the Latin square representing a uniformly random 1-factorisation of $K_{2n}$. Let $\ell(\mathcal{U})$ be the length of its longest row cycle, and let…

Let $S$ be a Steiner quasigroup of order $n$ selected uniformly at random. Let $\ell(S)$ be the length of its longest row cycle, and let $m(S)$ be the size of its largest proper St…

Let $\mathbf{A}$ be a non-negative Boltzmann half-pyramid, and let … be its height. Height stable-law conjecture. The random variable $H$ is in the domain of attraction of a stable…

For each $n$, independently include each symbol in each list $L_{ij}\subseteq\{1,\dots,n\}$ with probability $p$, and let $\mathcal{F}_{\mathrm{Latin}}$ be the property that there…

Conjecture on the maximum twin size. The logarithm of the maximum twin size, scaled by $\sqrt{\log n \cdot\log(\log n)}$, converges in probability to $2$:

For $k\leq n$, let $\mathcal{LR}_{k,n}$ be the set of $k\times n$ Latin rectangles with symbol set $[n]$. A partial Latin rectangle is $\beta n$-sparse if each row and column conta…

For $k\leq n$, let $\mathcal{LR}_{k,n}$ be the set of $k\times n$ Latin rectangles with symbol set $[n]$, and let $\randLR$ be chosen uniformly from this set. For every…

Let $G=H_n^d$ be the $d$-dimensional $n$-lattice, and let the deck consist of all its $d$-dimensional $k$-subgrids. A random $r$-colouring of $G$ is called reconstructible if it ca…