Steiner systems and configurations of points

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement

A note on the least number of edges of 3-uniform hypergraphs with upper chromatic number 2

AbstractThe upper chromatic number χ¯(H) of a hypergraph H=(X,E) is the maximum number k for which there exists a partition of X into non-empty subsets X=X1∪X2∪⋯∪Xk such that for each edge at least two vertices lie in one of the partite sets. We prove that for every n⩾3 there exists a 3-uniform hypergraph with n vertices, upper chromatic number 2 and ⌈n(n-2)/3⌉ edges which implies that a corresponding bound proved in [K. Diao, P. Zhao, H. Zhou, About the upper chromatic number of a co-hypergraph, Discrete Math. 220 (2000) 67–73] is best-possible

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

Good sequencings of partial Steiner systems

A partial (n, k, t)λ-system is a pair(X, B) where X is an n-set of vertices and B is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most λ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of {0,..., n − 1}. A sequencing is -block avoiding or, more briefly, -good if no block is contained in a set of vertices with consecutive labels. Here we give a short proof that, for fixed k, t and λ, any partial (n, k, t)λ-system has an -good sequencing for some = �(n1/t ) as n becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case k = t +1 where results of Kostochka, Mubayi and Verstraëte show that the value of cannot be increased beyond �((n log n)1/t ). A special case of our result shows that every partial Steiner triple system (partial (n, 3, 2)1- system) has an -good sequencing for each positive integer 0.0908 n1/2

Colouring 4-cycle systems with equitably coloured blocks

AbstractA colouring of a 4-cycle system (V,B) is a surjective mapping φ:V→Γ. The elements of Γ are colours and, for each i∈Γ, the set Ci=φ−1(i) is a colour class. If |Γ|=m, we have an m-colouring of (V,B). For every B∈B, let φ(B)={φ(x)|x∈B}. We say that a block B is equitably coloured if either |φ(B)∩Ci|=0 or |φ(B)∩Ci|=2 for every i∈Γ. Let F(n) be the set of integers m such that there exists an m-coloured 4-cycle system of order n with every block equitably coloured. We prove that: •minF(n)=3 for every n≡1(mod8), n⩾17, F(9)=∅,•{m|3⩽m⩽n+3116}⊆F(n), n≡1(mod16), n⩾17,•{m|3⩽m⩽n+2316}⊆F(n), n≡9(mod16), n⩾25,•for every sufficiently large n≡1(mod8), there is an integer m̄ such that maxF(n)⩽m̄. Moreover we show that maxF(n)=m̄ for infinite values of n

Computing techniques for the enumeration of cyclic Steiner systems

In this thesis a powerful algorithm is developed for finding cyclic Steiner systems. A cyclic Steiner system with parameters S(t,k,v) is a pair ( V,B), where B is a collection of subsets all of size k (called blocks) and V is a t; element set of points, such that each t-subset of V is contained in precisely one block of B. A Steiner system is called cyclic if it has an automorphism carrying the points in a v-cycle. The results obtained so far with this algorithm are given in Table VII of chapter 5. Among the values reported there, are the number of distinct cyclic solutions to S(2,3,55), S(2,3,57), S(2,3,61) and S(2,3,63) which are 121,098,240, 84,672,512, 2,542,203,904 and 1,782,918,144 respectively. These values were apparently unknown previous to this work