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Bollob\'as-type theorem determines the maximum cardinality of a Bollob\'as system of sets. The original result has been extended to various mathematical structures beyond sets, including vector spaces and affine spaces. This paper generalizes the Bollob\'as-type theorem to singular linear spaces, and determine the maximum cardinality of (skew) Bollob\'as systems on them.

In this paper we discuss a result similar to the polynomial version of the Alon-F\"uredi theorem. We prove that if you want to cover the vertices of the $n$-dimensional unit cube, except those of weight at most $r$ then you need an algebraic surface of degree at least $n-r$., Comment: 5 pages, published

There are many examples for point sets in finite geometry, which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod $p$, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behaviour of these renitent lines., Comment: The sharpness of some statements has been proved. There are some new examples and a new section about some dual statements and their consequences regarding the geometric properties of codewords of the $\mathbb{F}_p$-linear code generated by characteristic vectors of lines of $\mathrm{PG}(2,q)$. The latter result generalise an old result of Blokhuis, Brouwer and Wilbrink

In this article, we investigate collections of `well-spread-out' projective (and linear) subspaces. Projective $k$-subspaces in $\mathsf{PG}(d,\mathbb{F})$ are in `higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension $k$ in a generator set of points. We prove that the set $\mathcal{H}$ of higgledy-piggledy $k$-subspaces has to contain more than $\min{|\mathbb{F}|,\sum_{i=0}^k\lfloor\frac{d-k+i}{i+1}\rfloor}$ elements. We also prove that $\mathcal{H}$ has to contain more than $(k+1)\cdot(d-k)$ elements if the field $\mathbb{F}$ is algebraically closed. An $r$-uniform weak $(s,A)$ subspace design is a set of linear subspaces $H_1,..,H_N\le\mathbb{F}^m$ each of rank $r$ such that each linear subspace $W\le\mathbb{F}^m$ of rank $s$ meets at most $A$ among them. This subspace design is an $r$-uniform strong $(s,A)$ subspace design if $\sum_{i=1}^N\mathrm{rank}(H_i\cap W)\le A$ for $\forall W\le\mathbb{F}^m$ of rank $s$. We prove that if $m=r+s$ then the dual ($\{H_1^\bot,...,H_N^\bot\}$) of an $r$-uniform weak (strong) subspace design of parameter $(s,A)$ is an $s$-uniform weak (strong) subspace design of parameter $(r,A)$. We show the connection between uniform weak subspace designs and higgledy-piggledy subspaces proving that $A\ge\min{|\mathbb{F}|,\sum_{i=0}^{r-1}\lfloor\frac{s+i}{i+1}\rfloor}$ for $r$-uniform weak or strong $(s,A)$ subspace designs in $\mathbb{F}^{r+s}$. We show that the $r$-uniform strong $(s,r\cdot s+\binom{r}{2})$ subspace design constructed by Guruswami and Kopprty (based on multiplicity codes) has parameter $A=r\cdot s$ if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound $(k+1)\cdot(d-k)+1$ over algebraically closed field is tight., Comment: 27 pages. Submitted to Designs Codes and Cryptography

In this article we prove a theorem about the number of directions determined by less then $q$ affine points, similar to the result of Blokhuis et al. (in J. Comb. Theory Ser. A 86(1), 187-196, 1999) on the number of directions determined by $q$ affine points., Comment: 12 pages. NOTICE: this is the authors' version of a work that was submitted for publication in J. Algebr. Combin. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms are not reflected in this document. The final publication is available at Springer via doi:10.1007/s10801-012-0357-1

Let $U$ be a point set in the $n$-dimensional affine space ${\rm AG}(n,q)$ over the finite field of $q$ elements and $0\leq k\leq n-2$. In this paper we extend the definition of directions determined by $U$: a $k$-dimensional subspace $S_k$ at infinity is determined by $U$ if there is an affine $(k+1)$-dimensional subspace $T_{k+1}$ through $S_k$ such that $U\cap T_{k+1}$ spans $T_{k+1}$. We examine the extremal case $|U|=q^{n-1}$, and classify point sets NOT determining every $k$-subspace in certain cases., Comment: 7 pages. NOTICE: this is the author's version of a work that was accepted for publication in Discr. Math. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication

We examine sets of lines in PG(d,F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has more than 1.5d elements, and at least 2d-1 lines if the field F is algebraically closed. We show that suitable 2d-1 lines constitute such a set (if |F| > or = 2d-1), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong (s,A) subspace designs constructed by Guruswami and Kopparty have better (smaller) parameter A than one would think at first sight., Comment: 17 pages

Given a point set $U$ in an $n$-dimensional affine space of size $q^{n-1}-\varepsilon$, we obtain information on the structure of the set of directions that are not determined by $U$, and we describe an application in the theory of partial ovoids of certain partial geometries.

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 sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional space, are F_p-linear, hence confirming the linearity conjecture for blocking sets in this particular case.

Let Ck(n, q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n, q), q = p^h, p prime, h >= 1. In this pa- per, we show that there are no codewords of weight in the open interval ] q^{k+1}-1/q-1, 2q^k[ in Ck(n, q) \ Cn-k(n, q) which implies that there are no codewords with this weight in Ck(n, q) \ Ck(n, q) if k >= n/2. In par- ticular, for the code Cn-1(n, q) of points and hyperplanes of PG(n, q), we exclude all codewords in Cn-1(n, q) with weight in the open interval ] q^n-1/q-1, 2q^n-1[. This latter result implies a sharp bound on the weight of small weight codewords of Cn-1(n, q), a result which was previously only known for general dimension for q prime and q = p2, with p prime, p > 11, and in the case n = 2, for q = p^3, p >= 7 ([4],[5],[7],[8]).