23 problems

Let $G$ be a simple graph of order $n$, with adjacency eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$, and let $\lambda_h$ and $\lambda_\ell$ be its median eigenvalues, where ……

Let $\Gamma$ be a graph as in Hypothesis; in particular, $\Gamma$ is a strongly-regular graph in the setting of the paper. Complete classification conjecture. The graph $\Gamma$ is…

Extremal vector-space sunflower conjecture. There exists a constant $C=C(q,s)$ such that every $s$-sunflower-free family ${\mathcal F}$ of $k$-spaces over the field with $q$ elemen…

Let $t$ be a fixed positive integer. Let $\Omega_n$ be the set of all subspaces of an $n$-dimensional vector space over the field with $q$ elements, let $A_n^{(t)}$ be the family o…

Let $q$ be a prime power, let $\mathbb{F}_{q}^{n}$ be the $n$-dimensional vector space over $\mathbb{F}_{q}$, and call a family of its subspaces of density at least $\delta$ when i…

Let $\mathcal{L}_{n}(q)$ be the lattice of subspaces of $\mathbb{F}_{q}^{n}$, and let $\mathcal{Q}_{2}$ denote the diamond poset. Write $ex_{q}(n,\mathcal{Q}_{2})$ for the maximum…

Let $M$ be a simple $n$-element, rank-$r$ complex-representable matroid, where $r\fixed$ and $r73$. The average flat-size is the average of $|F|$ over the flats $F$ of $M$. Average…

Let $M$ be a simple $n$-element, rank-$4$, complex-representable matroid that is not the union of two lines. Its average plane-size is the average of $|P|$ over the rank-$3$ flats…

Let $V$ be an $n$-dimensional vector space over the finite field with $q$ elements, and let $\mathcal{L}(V)$ denote its lattice of subspaces. A family…

Let $\Sigma$ be a finite family of pairwise skew lines in $\mathbb{R}^4$. Suppose that the three-dimensional affine span (hull) containing any two of the lines contains at least tw…

Point-line incidence conjecture. If $G$ is $C_{2k}$-free, then

Let $[m]_q=(q^m-1)/(q-1)$ denote the $q$-analog of $m$. A graph is Singer-graceful when it satisfies the graceful-labeling condition associated with the Singer difference set. For…

Let $\mathbb{F}_0$ be a proper subfield of a field $\mathbb{F}$, and let $Q^+_{n-1}(2n-1,\mathbb{F})$ denote the polar Grassmannian of maximal singular subspaces of the hyperbolic…

Let $q$ be a prime power, and let $\mathrm{La}_q(n,\{Y_k,Y'_k\})$ denote the maximum size of a family of subspaces of an $n$-dimensional vector space over $\mathbb{F}_q$ containing…

The Weyl embedding conjecture. The Weyl embedding $\widehat{e}^W_n$ of $\widehat{\mathcal{Q}}_n$ induces on $\widetilde{\mathcal{Q}}_n$ its veronesean embedding $\widetilde{e}^W_n$…

Let $D$ be a design with the parameters of a geometric design $G$ in $\operatorname{PG}(n,q)$ or $\operatorname{AG}(n,q)$, where $q=p^m$. Let the $p$-rank denote the rank of the in…

Let $\Pi$ be a projective plane of order $p^s$, where $p$ is prime, and let its incidence matrix have rank over $\mathbb{F}_p$ called its $p$-rank. Hamada–Sachar conjecture. The…

Let $PG(r-1,q)$ be the projective geometry of rank $r$ over the finite field with $q$ elements, and let $C_p(PG(r-1,q))$ be the hypergraph whose hyperedges are the $p$-circuits of…

Round matroid hyperplane bound conjecture. If $M$ is round, has sufficiently large rank, and has no $U_{2,\ell+2}$-minor, then $M$ has at most

Let ${\cal B}$ be a biplane, let $a_1,a_2,\ldots,a_m\in{\cal \grave{M}}^v$ be vectors obeying the regular-vector property $|f_{a,j}^{-1}(2)|=q(a)$ for $j\in Q(a)$, and let … Here…

Let $\ell\ge 2$ be an integer, let $M$ be a rank-$r$ matroid, and let $U_{2,\ell+2}$ denote the rank-$2$ uniform matroid on $\ell+2$ elements. Let $q$ be the largest prime power su…

Let ${\mathscr C}$ be a clutter of a combinatorial affine plane, and suppose that its blocking number is at least $3$. An ideal solution clutter is an ideal clutter associated with…

Let $a(n)$ be the smallest multiple of the number of divisors of $n$ that is greater than or equal to $n$. On an $n\times n$ discrete torus, points are required to have no three in…