A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the (k−1)-dimensional projective space over Fq that have size at most cqk for some universal constant c. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of Fq-linear minimal codes of length n and dimension k, for every prime power q, for which n ≤ cqk. This solves one of the main open problems on minimal codes.

We study the problem of determining the minimum number of affine subspaces of codimension that are required to cover all points of at least times while covering the origin at most times. The case is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for. The value of also follows from a well-known theorem of Alon and FÜredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for we have, while for we have, and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.

Given a finite grid in R², how many lines are needed to cover all but one point at least k times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We solve this problem for various types of grids, in particular showing the tightness of the Ball–Serra bound when one side is much larger than the other. In other cases, we prove new lower bounds that improve upon Ball–Serra and provide an asymptotic answer for almost all grids. For the standard grid {0, …, n − 1} × {0, …, n − 1}, we prove nontrivial upper and lower bounds on the number of lines needed. To prove our results, we combine linear programming duality with some combinatorial arguments.

A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r−1 times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is t-intersecting, conjecturing that the cover number τ(H) of such a hypergraph H is at most r−t. In these papers, it was proven that the conjecture is true for r≤4t−1, but also that it need not be sharp; when r=5 and t=2, one has τ(H)≤2. We extend these results in two directions. First, for all t≥2 and r≤3t−1, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy τ(H)≤⌊(r−t)/2⌋+1. Second, we extend the range of t for which the conjecture is known to be true, showing that it holds for all [Formula presented]. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for k-wise t-intersecting hypergraphs, for all k≥3 and t≥1. We further give bounds on the cover numbers of strictly t-intersecting hypergraphs and the s-cover numbers of t-intersecting hypergraphs.