Note on a conjecture of sierksma

Abstract

Let S(q, d) be the maximal number v such that, for every general position linear map h: Δ(q-1)(d+1) →Rd, there exist at least v different collections {Δt1, ..., Δtq} of disjoint faces of Δ(q-1)(d+1) with the property that f(Δt1) ∩ ... ∩f(Δtq) ≠ Ø. Sierksma's conjecture is that S(q, d)=((q-1)!)d. The following lower bound (Theorem 1) is proved assuming that q is a prime number: {Mathematical expression} Using the same technique we obtain (Theorem 2) a lower bound for the number of different splittings of a "generic" necklace.