Shalom Eliahou

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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)
In this paper we solve, by computational means, an open problem of Erickson: denoting [n] = {1, . . . , n}, what is the smallest integer n0 such that, for every n ≥ n0 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)
The Cauchy-Davenport theorem states that, if p is prime and A,B are nonempty subsets of cardinality r, s in Z/pZ, the cardinality of the sumset A + B = {a + b | a ∈ A, b ∈ B} is bounded below by min(r+ s− 1, p); moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers r, s ≤ |G|,(More)
Let Dn 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 μDn (r, s) = min |A · B| of sumsets (product sets) A · B, where A and B range over all subsets of Dn of cardinality r and s respectively. It is shown by construction that μDn (r, s) is bounded above by the known value(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 dimK〈AB〉 and dimK〈B 〉 in terms of dimK A, dimK B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and(More)