A classical theorem of Kaiser and Rabinovich implies that if every subfamily of size at most $\lfloor\log_2(d+1)\rfloor+1$ of a finite family of compact connected sets in $\mathbb{R}^d$ has a common axis-parallel hyperplane transversal, then the entire family has an orthogonal star transversal, i.e., a union of $d$ hyperplanes, one perpendicular to each coordinate axis. We study colorful variants of this result. First, we use standard KKM machinery to establish a colorful analogue with $d+1$ color classes and a $\lfloor\log_2(d+1)\rfloor+1$ transversal hypothesis for rainbow subfamiles. Then we focus on reducing the required number of colors. To this end, we introduce the rainbow pushability $\pi_r(P)$ of a polytope and we use it to state and prove few-color variants of the colorful Carath\'eodory theorem and the colorful polytopal KKMS theorem. Applying the latter to the $d$-dimensional cube, we show that $\lfloor\log_2 d\rfloor+2$ color classes suffice for our transversal problem, provided that every rainbow choice has a common axis-parallel hyperplane transversal. We also extend both colorful results to the setting of convex $(n,D)$-bodies and we show that three colors are needed for $d\geq 2$.
Let $c>1$ be irrational. We prove that, for almost every $x>0$ and every $\varepsilon>0$, there are infinitely many triples $(p,r,s)$, with $p$ and $r$ prime and $s$ an integer, for which \[ 0<px-r<p^{-1/3+\varepsilon},\qquad 0<pcx-s<p^{-1/3+\varepsilon}. \] This improves the exponent $1/5$ in the homogeneous-line case...
Backman and Liu proved that every integral generalized permutohedron of type $A$, and in particular every matroid base polytope, admits a regular unimodular triangulation. The analogous statement fails in type $B$: the delta-matroid simplex \[\operatorname*{conv}\{\mathbf{0},\ e_1+e_2,\ e_1+e_3,\ e_2+e_3\}\] has normal...
Let $n,d\geq1$, and suppose that $E\subset \mathbb{Q}_p^n$ contains a degree-$d$ polynomial image of $\mathbb{Z}_p$ in every leading direction of a nonempty set $V\subset\mathbb{P}^{n-1}(\mathbb{Q}_p)$. We prove the sharp bound $\dim_H E\geq\dim_H V+1$. For joint block-valued polynomial evaluations, the total incidence...
Abstract polytopes and hypertopes are thin and residually-connected incidence geometries or, equivalently, chamber systems. Their degree of regularity is understood by looking at the action of their automorphism group on the set of their chambers. When this action has only one orbit, they are called regular. In this ar...
Claudio Alexandre Piedade, Philippe Tranchida· 0 citations
Let $I$ be the defining ideal of a general set of $s$ points in projective $N$-space over an algebraically closed field. Harbourne and Huneke conjectured the strengthened symbolic containment $$ I^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r $$ for every $r\ge 1$. For general points this was known in low dimension and for...
The canonical Ramsey theorem establishes that for every positive integer $t$, there is a sufficiently large $n$ such that every edge-colouring of a complete graph $K_n$ contains a copy of $K_t$ that is canonically coloured, i.e. monochromatic, rainbow, or lexicographic. In this paper we investigate two variations of th...
Strahinja Gvozdic, Zach Hunter, Aleksa Milojević et al.· 0 citations
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