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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