Related papers: Solution to Bishnoi's conjecture on minimal t-fold…

A $t$-fold blocking set of the finite Desarguesian plane $\mathrm{PG}(2,p^n)$, $p$ prime, is a set of points meeting each line of the plane in at least $t$ points. The minimum size of such sets is of interest for numerous reasons; however,…

We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of…

We present a construction for minimal blocking sets with respect to $(k-1)$-spaces in $\mathrm{PG}(n-1,q^t)$, the $(n-1)$-dimensional projective space over the finite field $\mathbb{F}_{q^t}$ of order $q^t$. The construction relies on the…

Blocking sets and minimal codes have been studied for many years in projective geometry and coding theory. In this paper, we provide a new lower bound on the size of $t$-fold $s$-blocking sets without the condition $t \leq q$, which is…

A blocking semioval is a set of points in a projective plane that is both a blocking set (i.e., every line meets the set, but the set contains no line) and a semioval (i.e., there is a unique tangent line at each point). The smallest size…

In this article we prove that if $(X,B+\beta)$ is a projective generalized klt pair such that $B+\beta$ is big, then $(X,B+\beta)$ admits a good Minimal Model or Mori fiber space. In particular, this implies Tossati's transcendental…

Let $n$ be a positive integer. Denote by $\mathrm{PG}(n,q)$ the $n$-dimensional projective space over the finite field $\mathbb{F}_q$ of order $q$. A blocking set in $\mathrm{PG}(n,q)$ is a set of points that has non-empty intersection with…

In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces $\operatorname{PG}(n,q)$ such that every plane is incident with at least one line of the set. This is the first main open problem…

In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking…

Blocking semiovals and the determination of their (minimum) sizes constitute one of the central research topics in finite projective geometry. In this article we introduce the concept of blocking set with the $r_\infty$-property in a finite…

Let $q$ be a fixed odd prime. We show that a finite subset $B$ of integers, not containing any perfect $q^{th}$ power, contains a $q^{th}$ power modulo almost every prime if and only if $B$ corresponds to a blocking set (with respect to…

A small minimal k-blocking set B in PG(n, q), q = pt, p prime, is a set of less than 3(qk + 1)/2 points in PG(n, q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies…

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space ${\rm PG}(3,q^3)$, one can find three $\mathbb F_q$-subgeometries such that the union of their point sets is a strong blocking set. This proves the…

A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen…

In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $<B>=PG(d,2)$ with $d$ odd and $B$ is a set of $d+2$ points of $PG(d,2)$ no $d+1$ of them in the same hyperplane. As a corollary to the…

The frame set conjecture for B-splines $B_n$, $n \ge 2$, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form $ab=r$, where $r$ is a rational number smaller than one and $a$…

In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking…

A skew Bollob\'{a}s system $\mathcal{P}=\{(A_i,B_i):1\leq i\leq m\}$ is a collection of pairs of disjoint subsets of $[n]$ such that $A_i\cap B_j\ne\emptyset$ for any $1\leq i<j\leq m$. Denote by $S_1(a, b)$ or $S_2(a, b)$ the maximum size…

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional…

We present a proof of the Harbourne-Hirschowitz conjecture for linear systems with base points of multiplicity seven or less. This proof uses a well-known degeneration of the projective plane, as well as a combinatorial technique that…