31 problems

Let $H$ be a graph. A graph $H$ forces quasirandomness if every graph sequence whose normalized homomorphism densities of $H$ converge to those of a quasirandom graph sequence is i…

Let $p$ be prime and define … Here $a$ is understood modulo $p$, $k\in\mathbb{Z}_p^\times$, and $M$ is in the range used for incomplete sums in the paper. Uniform power-sum conject…

Let $p$ be prime, let $\rho_{a,\tau}$ denote the exponentiation permutation defined using a primitive root $\tau$ modulo $p$, and write $D(\rho_{a,\tau})$ for its discrepancy. The…

Let $p$ be prime, let $\psi_k$ be the arithmetic permutation considered in the paper, and let $D(\psi_k)$ denote its discrepancy. Strong logarithmic discrepancy conjecture. … This…

Let $p$) be prime, let $\mathbb{Z}_p^\times$ denote the nonzero residue classes modulo $p$, and let $\psi_k$ be the arithmetic permutation defined in the surrounding discussion. Wr…

Existence conjecture. An $m$-symmetric permutation on sufficiently many symbols exists for every $m$, and one likely exists on $h(m)$ symbols.

Let $H=(V,E)$ be a $3$-graph. For $d otin[0,1]$ or $muleq 0$, no definition is needed here; for $din[0,1]$ and $mu0$, call $H$ uniformly $(d,mu)$-dense if every $Usubseteq V$ satis…

Let $F$ be a $3$-graph, and let $\pi_{\mathrm{U}}(F)$ denote its uniformly dense Turán density. In particular, $K^{(3)}_4$ is the complete $3$-graph on four vertices. Rödl's conjec…

Let $(\eta,\psi)$-quasirandom cylinder partitions be partitions of a tripartite $3$-graph satisfying the quasirandomness condition used in the cylinder regularity lemma, and let…

Minimum size conjecture for permutations. Every quasirandom-forcing set of permutations has cardinality at least six.

Minimum size conjecture for 4-point permutations. Every quasirandom-forcing set of $4$-point permutations has cardinality at least eight.

A linear combination of permutation patterns is quasirandom-forcing if it has the quasirandomness-forcing property studied in the paper. The number of terms means the number of per…

Let $\rho^*$ be the quasirandom-forcing linear combination of six permutations … A linear combination is quasirandom-forcing if it has the quasirandomness-forcing property studied…

Let $B$ be an oriented graph, let $\overline{B}$ be its underlying undirected graph obtained by forgetting edge directions, and call $B$ directed-forcing when it has the directed f…

Let $f\colon \mathbb{Z}/2^n\mathbb{Z}\to\{\pm1\}$ be a Boolean function. Say that $f$ has $(\delta,k)$-$\mathbb{R}$-regularity and that it has $\epsilon$-$\mathbb{Z}/2^n\mathbb{Z}$…

Unique inducibility conjecture. If all arities are at most $k$, then

A permutation is 3-symmetric if every subpermutation of length $3$ has the same density. A length is called admissible when it satisfies the divisibility constraints necessary for…

For a positive integer $r$, let $r(K_r)$ denote the Ramsey number of the complete graph $K_r$. An edge coloring of $K_N$ is $\theta$-quasirandom if, for every pair of disjoint vert…

Conlon–Fox–Sudakov forcing conjecture. Every bipartite graph containing a cycle has the forcing property.

Projective norm graph quasirandomness conjecture.

Higher-wise independence conjecture. For any prime power $q$ and integers $4\leq\ell<t$, all but $o(n^\ell)$ sets of $\ell$ vertices in $\operatorname{NG}(q,t)$ have…

Forcing conjecture. Every graph that is neither acyclic nor non-bipartite—that is, every graph that contains a cycle and is bipartite—is forcing.

Forcing conjecture. If the $H$-density of $G$ is $\alpha^{e(H)}+o(1)$, then $G$ is quasirandom with edge density $\alpha$.

Let $H$ be a graph, let $p\in[0,1]$, and let $G$ be a graph. The property $\mathcal{P}_{H,p}^{*}(\delta)$ means that every $U\subseteq V(G)$ satisfies … Let…

Let $H$ be a nonempty graph, let $p\nin[0,1]$, and let $G$ be a graph. For $\varepsilon>0$, write $\mathcal{P}_{H,p}^{*}(\varepsilon)$ for the property that every $U\subseteq V(G)$…