Note on a conjecture of sierksma

LetS(q, d) be the maximal numberv such that, for every general position linear maph: Δ^(q?1)(d+1) →R ^d, there exist at leastv different collections {Δ ^t1, ..., Δ ^t _q} of disjoint faces of Δ(q?1)(d+1) with the property thatf(Δ ^t1) ∩ ... ∩f(Δ ^t _q) ≠ Ø. Sierksma's conjecture is thatS(q, d)=((q?1)!) ^d . The following lower bound (Theorem 1) is proved assuming thatq is a prime number: $$S(q,d) \geqslant \frac{1}{{(q - 1)!}}\left( {\frac{q}{2}} \right)^{{{((q - 1)(d + 1))} \mathord{\left/ {\vphantom {{((q - 1)(d + 1))} 2}} \right. \kern-\nulldelimiterspace} 2}} .$$ Using the same technique we obtain (Theorem 2) a lower bound for the number of different splittings of a “generic” necklace.