80 problems

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices in which each $q$-set is present independently with probability $p$. A Steiner system is an…

Morris–Wang conjecture. The asymptotic results for $|\mathrm{Forb}(n,C_k^{(r)})|$ obtained for even $k$ and $r=3$ should hold for all $k,r\geq 3$.

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.

Let $H_k(n,m)$ be a uniformly random $k$-uniform hypergraph on $n$ vertices with $m$ hyperedges, and let $c$ denote the hyperedge-to-vertex ratio. A hypergraph is 2-colorable if it…

Let a heterogeneous random hypergraph contain hyperedges of multiple degrees, and suppose its projected graph is observed under the density condition studied by Morgan et al. Their…

Let $P_M^r$ and $P_M^s$ denote the distributions of a measure $M$ for random nodes and hyperedges, respectively, and let $\delta(P_M)=\delta(P_M^r,P_M^s)$. For simplices, write…

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 $\mathcal{G}^{(q)}(n,p)$ 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…

Liu--Zhao's conjecture.

Let $r,q\in\mathbb{N}$ and let $\varepsilon,\mu>0$. Consider the random $r$-uniform hypergraph at edge probability $p=Cn^{-r+1}\log n$, together with the conclusion of Theorem in t…

Exact-recovery threshold conjecture. For $d=4$ and $5$, the correct threshold for exact recovery is $\frac{2d-4}{2d-1}$. The paper establishes only bounds on this threshold: exact…

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…

In the random connection hypergraph model, let $gamma$ and $gamma'$ denote the model parameters governing the relevant vertex-mark distributions, and let the triangle count denote…

In the random connection hypergraph model, let $gamma$ and $gamma'$ denote the model parameters governing the relevant vertex-mark distributions, and let the Betti numbers denote t…

Let $r$-uniform Erdős–Rényi random hypergraphs on $n$ vertices have average degree $d$. The maximum independent-set density is asymptotically…

Kang–Kelly–Kühn–Methuku–Osthus conjecture. If

Let $\tilde K_t$ denote the $3$-uniform linear clique of order $t$, let $\mathbb{H}(n,p)$ denote the binomial random $3$-uniform hypergraph, and let…

For a random $d$-regular $k$-uniform hypergraph, let $d_\star(k)$ denote the explicit upper bound on the satisfiability, equivalently $2$-colorability, threshold established in the…

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

Growth-rate conjecture for random 3-uniform linear 4-cycles. For $n^{-1}\leq p\leq n^{-1/2}$, the correct growth rate of $\mathrm{ex}(G_{n,p}^{(3)},C_4^{(3)})$ is $n^{4/3}$.

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…

Random-hypergraph conjecture. Using a random hypergraph model, this proportion should be

Marginal-velocity conjecture. In line with previous results, the selected value of $\theta$ is the one for which

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…