28 problems

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices, and let an $(n,q,r)$-Steiner system be a collection of $q$-sets containing every $r$-set exactly…

Let $G_r(n,p)$ be the random $r$-uniform hypergraph, and let $C^{(r)}_{n,\ell}$ be the $r$-uniform $\ell$-cycle on $n$ vertices. Write $Z(C^{(r)}_{n,\ell})$ for the number of copie…

Piecewise asymptotic conjecture for random 3-uniform linear 4-cycles.

Let $G^3_{n,p}$ be the random $3$-uniform hypergraph on $n$ vertices in which each $3$-edge is present independently with probability $p$, and let $C^3_4$ be the $3$-uniform linear…

Asymptotic clique-count conjecture. The expected number of cliques satisfies

Let $t,k,\lambda\in\mathbb N$ with $k>t$, let $V$ be an $n$-element set satisfying the divisibility conditions $\binom{k-i}{t-i}\mid\lambda\binom{n-i}{t-i}$ for every…

Let $n\equiv 1,3 \pmod 6$, and let $T_1,T_2,\ldots$ be a uniformly random ordering of the triples in $\binom{[n]}{3}$. Steiner triple system hitting-time conjecture. With high prob…

Łuczak–Peled conjecture. For every $d \geq 2$ and $p=p(n)$ such that $|np-c_d|$ is bounded away from $0$, $H_{d-1}(Y_d(n,p))$ is torsion-free asymptotically almost surely.

Kamčev–Liebenau–Wormald conjecture. There exists a set $\mathfrak{W}$ having probability $1-O(n^{-\omega(1)})$ in both $\mathcal{D}_r(n,m)$ and $\mathcal{T}_r(n,m)$ such that, unif…

Let $G_{n,p}^{(3)}$ be the random 3-uniform hypergraph on $n$ vertices, let $C_4^{(3)}$ be the 3-uniform expansion of the 4-cycle, and let…

Random transference conjecture. Asymptotically almost surely, every spanning subgraph $G'\subseteq G$ satisfying

Interpolation conjecture. For $\frac{3}{2}<x\leq 4$,

Stable-labeling variational asymptotic conjecture. If $n\to\infty$ and $p=p(n)$ satisfies $n^{-1/\Delta}\ll p\ll 1$, then

Clique-and-hub conjecture. The variational problem is asymptotically optimized by planting a clique and a hub.

Constant discrepancy conjecture. There is an absolute constant $K>0$ such that, for every integer $t>0$, a random $t$-regular hypergraph $\mathcal{H}$ on $n$ vertices and…

Let $G^{(r)}(n,p)$ be the binomial random $r$-uniform hypergraph. For $r\geq 3$, define … This is the sharp threshold for the disappearance of isolated vertices. Degree-threshold c…

Let $G^{(r)}(n,p)$ be the binomial random $r$-uniform hypergraph, and let $X_\ell$ count its Hamiltonian $\ell$-cycles. Expectation-threshold conjecture. For all integers…

Let $\mathcal{H}^k(n,p)$ be the random $k$-uniform hypergraph with edge probability $p$, and let $j$-components denote components under the adjacency notion used in the paper. Writ…

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, and expose its edges one by one in a uniformly random order to obtain the random hypergraph process. A facet is…

Fractional-decomposition threshold conjecture. The threshold for the appearance of fractional $K_k$-decompositions in $\mathcal{H}_k(n;p)$ is

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, and expose its edges one by one in a uniformly random order to obtain the random hypergraph process. A facet is…

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, let $\mathcal{H}_k(n;p)$ be the random $k$-uniform hypergraph obtained by retaining each edge independently with…

Packing conjecture. There exists a constant $C$ such that, if $(k-l)\mid n$ and

Let ${\cal H}$ be the random $k$-uniform hypergraph with edge-probability $\varphi$, let $\Delta$ denote its maximum degree, and let $\Lambda'(\Delta)$ be the parameter from Theore…

Let $\mathbb H^{(k)}(n,d)$ denote the random $d$-regular $k$-uniform hypergraph on $n$ vertices. For integers $k>\ell\geq 2$, an $\ell$-overlapping cycle is a $k$-uniform hypergrap…