Minimum number of additive tuples in groups of prime order For a prime number p and a sequence of integers a0, . . . , akxe2x88x88 0,1, . . . , p, lets (a0, . . . , ak) be the minimum number of (k 1)tuples (x0, . . . , xk) xe2x88x88A0xc3x97xc2xb7xc2xb7xc2xb7xc3x97Akwithx0x1xc2xb7xc2xb7xc2xb7xk, over subsets a0, . . . , Akxe2x8ax86Zp of sizes a0, . . . , ak respectively. We observe that an elegant argument of Samotij and Sudakov can be extended to show that there exists an extremal configuration with all sets Ai being intervals of appropriate length. The same conclusion also holds for the related problem ,posed by Bajnok, whena0xc2xb7xc2xb7xc2xb7ak:aandA0xc2xb7xc2xb7xc2xb7Ak, provided k is not equal 1 modulop. Finally, by applying basic Fourier analysis, we show for Bajnokxe2x80x99s problem that if pu003e13 and axe2x88x88 3, . . . , pxe2x88x923are fixed whilekxe2x89xa11 (modp) tends to infinity, then the extremal configuration alternates between at least two affine nonequivalent sets.

On the cardinality spectrum and the number of latin bitrades of order 3 By a (Latin) unitrade of order k, we call a subset of vertices of the Hamming graph H(n, k) that intersects any maximal clique at either 0 or 2 vertices. A bitrade is a bipartite unitrade, i.e., a unitrade that can be split into two independent subsets. We study the cardinality spectrum of bitrades in the Hamming graph H(n, k) with k  3 (ternary hypercube) and the growth of the number of such bitrades as n grows. In particular, we determine all possible small (up to 2.5xc2xb72n) and large (from 14xc2xb73nxe2x88x923) cardinalities of bitrades of dimension n and prove that the cardinality of a bitrade takes only values equivalent to 0 or 2n modulo 3 (this result can be treated in terms of a ternary ReedMuller type code). A part of the results are valid for an arbitrary k. For k  3 and n xe2x86x92 xe2x88x9e we prove that the number of nonequivalent bitrades is not less than 2(23xe2x88x92o(1))n and not greater than 2alphan, xcexb1 u003c 2 (the growth order of the double logarithm of this number remains unknown). Alternatively, the studied set Bn of bitrades of order 3 can be defined as follows: B0 consists of three rationals  1, 0, 1; Bn consists of ordered triples (a, b, c) of elements from Bnxe2x88x921 such that a  b  c  0.