43 problems

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Let $\genfrac{[}{]}{0pt}{}{V}{1}_q$ denote the family of $1$-dimensional subspaces of $V$, and let…

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…

Let $k\in 3\mathbb{N}$, let $(\mathbb{Z}/k\mathbb{Z})^2$ be the discrete torus, and let $A\subseteq(\mathbb{Z}/k\mathbb{Z})^2$. The lines $L$ are the axis-parallel or diagonal line…

Prime-power conjecture for projective planes. If a projective plane of order $n$ exists, then $n$ must be a prime power.

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…

Biplane finiteness conjecture. Only finitely many biplanes exist.

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 $q>1$, let $\Omega_n$ be the set of all subspaces of an $n$-dimensional vector space over the field with $q$ elements, and let $\mu_{\sigma}$ be the probability measure on…

Ovoid fix-structure conjecture. No such ovoid is the fix structure of any collineation.

Prime Power conjecture. When $k$ is not a prime power, there is no finite affine plane with $k^2$ vertices.

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 $M$ be a simple $n$-element, rank-$3$, complex-representable matroid. Its average line-length is the average of $|L|$ over the rank-$2$ flats $L$ of $M$. Average line-length co…

Let $Gr(V)$ be the Grassmannian poset with boundary operator $partial^{(R)}$ on chains with coefficients in a ring $R$, and suppose that $\operatorname{char}(R)$ is coprime to $q$.…

Bérczi–Schwarcz–Yamaguchi's conjecture. Every matroid is $(1,2)$-decomposable.

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…

Attainability conjecture. The upper bound from Theorem, and the corresponding bound for Ferrers diagram rank-metric codes, can always be attained.

Consider partitions of $H(5,5)$ into $25$ $(5,5^3,3)_5$ MDS codes, and call a partition lengthenable if it can be extended to a partition of $H(6,5)$ into $25$ perfect codes. Scarc…

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…

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