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

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

On colorful edge triples in edge-colored complete graphs

An edge-coloring of the complete graph Kn we call F-caring if it leaves no F-subgraph of Kn monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when F=K1,3 and F=P4 we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring of Kn. An explicit family of 2⌈log2n⌉ 3-edge-colorings of Kn so that every quadruple of its vertices contains a totally multicolored P4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the five length cycle