Steiner configurations ideals: Containment and colouring

Given a homogeneous ideal I ⊆ k[x0, …, xn ], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m, r ∈ N, I(m) ⊆ Ir holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in Pnk. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph H, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H. We apply these results in the case that H is a Steiner System

Blocking sets and colouring in Steiner systems S(2,4,ν)

A Steiner system S(2, 4, v) is a v-element set V together with a collection B of 4-subsets of V called blocks such that every 2-subset of V is contained in exactly one block

strongly balanced 4 kite designs nested into oq systems

In this paper we determine the spectrum for octagon quadrangle systems [OQS] which can be partitioned into two strongly balanced 4-kitedesigns

Chromatic Polynomials of Mixed Hypercyles

We color the vertices of each of the edges of a C-hypergraph (or cohypergraph) in such a way that at least two vertices receive the same color and in every proper coloring of a B-hypergraph (or bihypergraph), we forbid the cases when the vertices of any of its edges are colored with the same color (monochromatic) or when they are all colored with distinct colors (rainbow). In this paper, we determined explicit formulae for the chromatic polynomials of C-hypercycles and B-hypercycles