We study the problem of finding the largest numberT(n,m) of ternary vectors of lengthnsuch that for any three distinct vectors there are at leastmcoordinates where they pairwise differ. This problem is a special case of the perfectk-hashing problem in theoretical computer science, corresponding to thek= 3 case. Form= 1, we get the classical trifference problem which is wide open. We prove upper and lower bounds onT(n,m) for various ranges of the parametermand determine the phase transition threshold onm=m(n) whereT(n,m) jumps from constant to exponential in n. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters. Moreover, we relate the trifference problem to the sunflower conjecture.

Abstract In this paper we study the number of special directions of sets of cardinality divisible by p on a finite plane of order p, where p is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly three special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that have exactly four special directions for small values of p.