Related papers: Planar posets have dimension at most linear in the…
We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $H$ is a family in $B_n$ with $|H|\ge (q-1+\varepsilon){n\choose…
We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad…
We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2…
When we have a morphism f : P^n -> P^n, then we have an inequality \frac{1}{\deg f} h(f(P)) +C > h(P) which provides a good upper bound of $h(P)$. However, if $f$ is a rational map, then \frac{1}{\deg f} h(f(P))+C cannot be an upper bound…
In this paper we prove that for all pairs $(d,m)$ with $d/m \geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension.
We investigate structural properties of the cone of roots of relative Steiner polynomials of convex bodies. We prove that they are closed, monotonous with respect to the dimension, and that they cover the whole upper half-plane, except the…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small…
A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some…
We prove Conjecture 4.16 of the paper [EL21] of Elagin and Lunts; namely, that a smooth projective curve of genus at least 1 over a field has diagonal dimension 2.
Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-$n$ poset, is to compute its number of linear extensions. In 1991…
We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m-dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6,s if the…
We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e., projective systems. It is known that the maximum number of $(h-1)$-spaces in PG$(2,q)$, such that no hyperplane…
We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension…
The queue-number of a poset is the queue-number of its cover graph viewed as a directed acyclic graph, i.e., when the vertex order must be a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has…
We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our…
We prove a new upper bound on the essential p-dimension of the projective linear group PGLn.
A (Hasse) diagram of a finite partially ordered set (poset) P will be called quasiplanar if for any two incomparable elements u and v, either v is on the left of all maximal chains containing u, or v is on the right of all these chains.…
(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an…
We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.