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A matching in a group G is a bijection ϕ from a subset A to a subset B in G such that aϕ(a) / ∈ A for all a ∈ A. The group G is said to have the matching property if, for any finite subsets A, B in G of same cardinality with 1 / ∈ B, there is a matching from A to B. Using tools from additive number theory, Losonczy proved a few years ago that the only(More)
Let D n be the dihedral group of order 2n. For all integers r, s such that 1 ≤ r, s ≤ 2n, we give an explicit upper bound for the minimal size µ D n (r, s) = min |A · B| of sumsets (product sets) A · B, where A and B range over all subsets of D n of cardinality r and s respectively. It is shown by construction that µ D n (r, s) is bounded above by the known(More)
In this paper we solve, by computational means, an open problem of Erickson: denoting [n] = {1,. .. , n}, what is the smallest integer n 0 such that, for every n ≥ n 0 and every 2-coloring of the grid [n] × [n], there is a constant 2-square, i.e. a 2 × 2 subgrid S = {i, i + t} × {j, j + t} whose four points are colored the same? It has been shown recently(More)
Let K ⊂ L be a field extension. Given K-subspaces A, B of L, we study the subspace AB spanned by the product set AB = {ab | a ∈ A, b ∈ B}. We obtain some lower bounds on dim K AB and dim K B n in terms of dim K A, dim K B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman(More)