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Capturing forms in dense subsets of finite fields

Brandon Hanson (2013)

Acta Arithmetica

An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field q . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size of A ⊂ q to guarantee...

Cardinality of height function’s range in case of maximally many rectangular islands - computed by cuts

Eszter Horváth, Branimir Šešelja, Andreja Tepavčević (2013)

Open Mathematics

We deal with rectangular m×n boards of square cells, using the cut technics of the height function. We investigate combinatorial properties of this function, and in particular we give lower and upper bounds for the number of essentially different cuts. This number turns out to be the cardinality of the height function’s range, in case the height function has maximally many rectangular islands.

Codes that attain minimum distance in every possible direction

Gyula Katona, Attila Sali, Klaus-Dieter Schewe (2008)

Open Mathematics

The following problem motivated by investigation of databases is studied. Let 𝒞 be a q-ary code of length n with the properties that 𝒞 has minimum distance at least n − k + 1, and for any set of k − 1 coordinates there exist two codewords that agree exactly there. Let f(q, k)be the maximum n for which such a code exists. f(q, k)is bounded by linear functions of k and q, and the exact values for special k and qare determined.

Combinatorics of open covers (VII): Groupability

Ljubiša D. R. Kočinac, Marion Scheepers (2003)

Fundamenta Mathematicae

We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a T 31 / 2 -space. In [9] we showed that C p ( X ) has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. C p ( X ) has countable fan tightness and the Reznichenko property. 2....

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