NOTE ON A CONJECTURE OF SIERKSMA

Let S(q, d) be the maximal number v such that, for every general posit ion linear map h: DELTA(q-1)(d+1) --> R(d), there exist at least v dif ferent collections {DELTA(t1), ..., DELTA(t)q} of disjoint faces of DE LTA(q-1)(d+1) with the property that f(DELTA(t1)) and...and f(DELTA(t) q) not-equal empty set. 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: S(q,d) greater-than-or-equal-to 1/(q-1)! (q/2)((q-1 )(d+1))/2 Using the same technique we obtain (Theorem 2) a lower bound for the number of different splittings of a ''generic'' necklace.