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a b s t r a c tA vector space partition P of a finite dimensional vector space V = V (n, q) of dimension n over a finite field with q elements, is a collection of subspaces U 1 , U 2 , . . . , U t with the property that every non zero vector of V is contained in exactly one of these subspaces.The tail of P consists of the subspaces of least dimension d 1 in P , and the length n 1 of the tail is the number of subspaces in the tail. Let d 2 denote the second least dimension in P .Two cases are considered: the integer q d 2 −d 1 does not divide respective divides n 1 . In the first case it is proved that if 2d 1 > d 2 then n 1 ≥ q d 1 + 1 and if 2d 1 ≤ d 2 then either n 1 = (q d 2 − 1)/(q d 1 − 1) or n 1 > 2q d 2 −d 1 . These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2d 1 respectively d 2 .In case q d 2 −d 1 divides n 1 it is shown that if d 2 < 2d 1 then n 1 ≥ q d 2 − q d 1 + q d 2 −d 1 and if 2d 1 ≤ d 2 then n 1 ≥ q d 2 . The last bound is also shown to be tight.The results considerably improve earlier found lower bounds on the length of the tail.

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A vector space partition is here a collection P of subspaces of a finite vector space V (n, q), of dimension n over a finite field with q elements, with the property that every non zero vector is contained in a unique member of P. Vector space partitions relates to finite projective planes, design theory and error correcting codes.In the first part of the talk I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the talk contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V (n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition.Finally, I will also give a few historical remarks.