Related papers: Large $(k; r, s; n, q)$-sets in Projective Spaces

In this paper we present and analyze computational results concerning small complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$ and $N=4$ over the finite field of order $q$. The results have been obtained using…

The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{\alpha, -\alpha\}$,…

There has been much work on the following question: given n how large can a subset of {1,...,n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch…

Let G=PGL(2,q) be the projective general linear group acting on the projective line P_q. A subset S of G is intersecting if for any pair of permutations \pi,\sigma in S, there is a projective point p in P_q such that p^\pi=p^\sigma. We…

Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and…

We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in…

We construct two-weight sets in PG$(3n-1,q)$, $n\geq2$ with the same weights as those that would arise from the blow-up of a maximal $q$-arc in PG$(2,q^n)$. The construction is of particular interest when $q$ is odd, as it is well known…

The main purpose of this paper is to find double blocking sets in $\mathrm{PG}(2,q)$ of size less than $3q$, in particular when $q$ is prime. To this end, we study double blocking sets in $\mathrm{PG}(2,q)$ of size $3q-1$ admitting at least…

The length function $\ell_q(r,R)$ is the smallest possible length $n$ of a $ q $-ary linear $[n,n-r]_qR$ code with codimension (redundancy) $r$ and covering radius $R$. Let $s_q(N,\rho)$ be the smallest size of a $\rho$-saturating set in…

An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set such that each partition has $k$ nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an…

In this work complete caps in $PG(N,q)$ of size $O(q^{\frac{N-1}{2}}\log^{300} q)$ are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound $\sqrt{2}q^{\frac{N-1}{2}}$ and it…

In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\H{o}s conjectured in 1990 that the number of such…

In this work, we prove the existence of maximal partial line spreads in PG(5,q) of size q^3+q^2+kq+1, with 1 \leq k \leq (q^3-q^2)/(q+1), k an integer. Moreover, by a computer search, we do this for larger values of k, for q \leq 7. Again…

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…

Let $S \subset \R^{k + m}$ be a compact semi-algebraic set defined by a system of $\ell$ polynomial inequalities of degree at most 2. $ Let $\pi$ denote the standard projection from $\R^{k + m}$ onto $\R^m$. We prove that for any $q >0$,…

We prove new formulas and congruences for $p(n,k):=$ the number of partitions of $n$ into $k$ parts and $q(n,k):=$ the number of partitions of $n$ into $k$ distinct parts. Also, we give lower and upper bounds for the density of the set…

Asymptotics are given for the number of rational points in the domain of a morphism of weighted projective stacks whose images have bounded height and satisfy a (possibly infinite) set of local conditions. As a consequence we obtain results…

In the affine space $\mathbb{F}_q^n$ over the finite field of order $q$, a point set $S$ is said to be $(d,k,r)$-evasive if the intersection between $S$ and any variety, of dimension $k$ and degree at most $d$, has cardinality less than…

A symmetric subset of the reals is one that remains invariant under some reflection x --> c-x. Given 0 < x < 1, there exists a real number D(x) with the following property: if 0 < d < D(x), then every subset of [0,1] with measure x contains…

For nonempty sets $A,B$ of nonnegative integers and an integer $n$, let $r_{A,B}(n)$ be the number of representations of $n$ as $a+b$ and $d_{A,B}(n)$ be the number of representations of $n$ as $a-b$, where $a\in A, b\in B$. In this paper,…