In this paper we study irreducible simplicial arrangements of projective planes in $\mathbb{P}^{3}(\mathbb{R})$ from combinatorial and projective geometry viewpoints. We first formulate a simpliciality criterion in terms of incidences between rank-two and rank-three flats, together with equivalent formulations using face numbers, reduced restrictions, and characteristic polynomials. We also relate the classical planar restriction data of Gr\"unbaum-Shephard to Ziegler multirestrictions. Our principal result concerns the rank-four special-vertex property: among the irreducible crystallographic Coxeter arrangements of rank four, the arrangements of types $A_4$ and $B_4$ admit a special vertex, whereas those of types $D_4$ and $F_4$ do not. A simplicial deletion chain inside $B_4$ supplies further irreducible examples with a special vertex. Finally, we compare rank-flat, Purdy-type, and Gr\"unbaum-Shephard defects for these arrangements.
We prove that every simplicial triangulation of real projective $d$-space has $\exp(\Omega(\sqrt d))$ vertices. Together with known constructions, this determines the minimum vertex number as $\mu_d=\exp(d^{1/2+o(1)})$. The result follows from a topological generalization of the Figiel--Lindenstrauss--Milman inequality...
Florian Frick, Kaave Hosseini, Eric Myzelev et al.· 1 citation
We study regular unimodular triangulations, the integer decomposition property, and Ehrhart-theoretic properties of two families of Hermite normal form simplices. We first consider the one-row case associated with the vector $(N - 1, \dots ,N - 1 , N)\in \mathbb{N}^d$, and completely characterize when the corresponding...
Justus Bruckamp, Jhon B. Caicedo, Martina Juhnke· 1 citation· ⚡1
We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$...
A collection of $m \geq 2$ lines through the origin in $\mathbb{R}^2$ is balanced if the sum of the orthogonal projections onto these lines equals $m/2$ times the identity. A line arrangement in $\mathbb{RP}^2$, endowed with the round metric of curvature $1$, is point-balanced if the tangent lines at every vertex form...
We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $\Delta_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $\Delta_o$ together with its barycenter. As...
For an ordered ideal triangulation of a knot exterior in $S^3$, we give a cochain construction relating its face matrix $C$ to its preferred-longitude gluing matrices $A,B$. The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix id...
Sai Kambampati· 0 citations
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