105 problems

Mihail–Vazirani conjecture. Every $0/1$ polytope $P$ satisfies

Let $\mathcal{G}$ be a fixed $k$-regular graph and let $\mathcal{G}_p$ be a random lift of degree $p$. Denote by $\lambda_{\mathrm{new}}(\mathcal{G}_p)$ the maximum absolute value…

Lubotzky–Sarnak conjecture. For some sequence of finite-index normal subgroups intersecting only at the identity, the Cayley graphs of the corresponding finite quotients do not for…

For $varepsilon>0$, let an $(\varepsilon,d)$-expander be the sublinear-expansion notion used in the source. Hamiltonicity conjecture for regular sublinear expanders. There exists…

Let the graphs arising from the other surfaces in the paper's family be the analogous graphs obtained from the corresponding surface actions. Bourgain–Gamburd–Sarnak's expander con…

Let $G$ be a nonabelian finite simple group. A subset $S\subset G$ is symmetric when it is closed under taking inverses, and $\operatorname{Cay}(G,S)$ denotes the Cayley graph of…

Let $(G_i)$ be the family of all non-abelian finite simple groups. A Cayley graph $\mathcal{C}(G_i,S_i)$ is the graph associated with the generating set $S_i$, and a family of grap…

Cheeger-constant conjecture. There exists a constant $c$ such that

Let $Sym(n)$ act on the set $X_{n,4}$ of $4$-tuples of $n$ elements. For a generating set $S$, write $Cay(Sym(n);S)$ for the corresponding Cayley graph and $Sch(X_{n,4};S)$ for the…

Let $\mathcal{F}_n$ be the free group of rank $n$, and let $\operatorname{Aut}(\mathcal{F}_n)$ and $\operatorname{Out}(\mathcal{F}_n)$ denote its automorphism and outer automorphis…

Let $G_i$ be the groups in an expander family of Cayley graphs $\mathcal{C}(G_i,S_i)$, and let $\operatorname{Aut}(F_n)$ denote the automorphism group of a free group. Property tau…

Let $\Gamma_i$ be a family of finite groups, and let $\Sigma_i$ and $\Sigma_i'$ be generating sets of $\Gamma_i$ whose sizes are uniformly bounded. Lubotzky–Weiss generating-set co…

Let $d$ be sufficiently large, and let a $d$-regular expander graph be defined through the spectral expansion condition considered by Krivelevich and Sudakov. Krivelevich–Sudakov c…

Let $f\in\mathbb R[x,y]$ be neither additive nor multiplicative. For a finite set $A\subset\mathbb R$, write $f(A,A)=\{f(a_1,a_2):a_1,a_2\in A\}$. Near-quadratic Elekes–Rónyai expa…

Let $G$ be an $n$-vertex $d$-regular graph with eigenvalues $d=\lambda_1(G)\geq\lambda_2(G)\geq\dots\geq\lambda_n(G)$, and call $G$ an $(n,d,\lambda)$-graph when…

Fix $t$, and let $n$ be large enough that the parameters $m_2,\ldots,m_t$ defined by … where $c\in(-1,1)$ is chosen so that $\sum_i m_i=n+1$. Let $H$ be the Tanner-code parity-chec…

Fix constants $0<\alpha<1$ and $c_1,c_2>0$, and let $\epsilon>0$ be sufficiently small. Let $C_1=C_1(\epsilon,\alpha,c_1,c_2)>0$ and $C_2=C_2(\epsilon,\alpha,c_1,c_2)>0$ be suitabl…

Let $(\mathcal{G}_{n})_{n}$ be the family of $\operatorname{SL}(2,\mathbb{Z})$-orbit graphs of primitive $n$-squared origamis in the stratum $\mathcal{H}(2)$. McMullen's expander c…

Fix a sequence $(g_n)_{n\geq0}$ such that … Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ on $3n$ edges, and let $\mathbf{T}_{2n,g_n}^*$ be its dual map. For…

A high genus map is a combinatorial map obtained by gluing polygons along their edges, with both its genus and size tending to infinity. An expander subgraph is a subgraph with uni…

Let $\mathbb{P}_m(n)$ be the undirected prefix-reversal graph, which is $2n$-regular for $m>1$. A sequence of regular graphs is an expander family if there exists $\varepsilon>0$ s…

A primitive origami is a square-tiled surface whose covering of the unit square torus does not factor through a nontrivial isogeny. The associated $\operatorname{SL}(2,\mathbb{Z})$…

Let $E \subseteq \mathbb{F}_q^d$ and let $R \subseteq \mathbb{F}_q \setminus \{0\}$. Let $C$ be the constant appearing in Theorem, whose conclusion guarantees the existence of the…

Let $D\in\mathbb{N}$. An $(n,d,\lambda)$-graph is a graph on $n$ vertices with degree $d$ and second eigenvalue bounded by $\lambda$. Let $T$ be a tree of maximum degree at most…

For a translation surface in the stratum $\mathcal{H}(2)$, the associated Veech group is arithmetic when its orbit gives rise to the graphs under consideration. McMullen's expansio…