Bernt Lindström

Determination of two vectors from the sum

Abstract Let S m be the set of all vectors of dimension m with all components 0 or 1. Let g4(m) be the maximum of | A + B | for pairs A, B of subsets of S m such that the sums a+b are different for different pairs ( a, b ), a ∈ A , b ∈ B . Let λ(m) be the maximum of | A |, A ⊃ S m , such that the sums a 1 + a 2 are different for different subsets { a 1 , a 2 } in A . Let ν(m) be the maximum of | A |, A ⊃ S m , for A such that the sums a 1 + a 2 are different modulo 2 for different subsets { a 1 , a 2 } in A , a 1 ≠ a 2 . The problem is to estimate φ(m) 1/ m , λ(m) 1/ m and ν(m) 1/ m as m →∞.

On B2-sequences of vectors

Abstract We shall prove a generalization of a classic inequality of Erdos and Turan on B 2 -sequences of integers to B 2 -sequences of vectors with integer components. We improve a recent result by the author when the dimension tends to infinity.

A Partition of L(3, n) into Saturated Symmetric Chains

For positive integers m and n let L ( m , n ) denote the set of all m -tuples ( a 1 , a 2 ..., a m ) of integers with 0≤ a 1 ≤ a 2 ⋯≤ a m ≤ n . The set L ( m , n ) is partially ordered such that ( a 1 ,..., a m )≤( b 1 ,... b m ) holds precisely when a i ≤ b i for i = 1, 2,..., m . We prove that the partially ordered set L (3, n ) has a partition into saturated symmetric chains. It is not out of the place to mention that D. E. Littlewood assumed that there is such a partition of L ( m , n ) into symmetric chains for all m , n ≥ 1 in his book Theory of Group Characters .

A theorem on families of sets

Abstract We prove the following result and transfinite extensions of it: Let (Mi:i ϵ I) be a family of non-zero subsets of the set S. If the cardinalities |I| = f and |S| = n are finite and f > n(r − 1), then one can find r disjoint subsets Iυ(υ = 1,…,r) of I for which ⋃ i∈I 1 M i = … = ⋃ i∈I r M i The proof is constructive. We apply a generalization by R. Rado of P. Halls celebrated theorem on systems of representatives. Another proof of the above result has been found by H. Tverberg (see [3]). Tverberg applies his generalization of Radons theorem (see [2]). He also shows by an example that the result is in a sense best possible.

A Desarguesian theorem for algebraic combinatorial geometries

The points of an algebraic combinatorial geometry are equivalence classes of transcendentals over a fieldk; two transcendentals represent the same point when they are algebraically dependent overk. The points of an algebraically closed field of transcendence degree two (three) overk are the lines (resp. planes) of the geometry.We give a necessary and sufficient condition for two coplanar lines to meet in a point (Theorem 1) and prove the converse of Desargues’ theorem for these geometries (Theorem 2). A corollary: the “non-Desargues” matroid is non-algebraic.The proofs depend on five properties (or postulates). The fifth of these is a deep property first proved by Ingleton and Main [3] in their paper showing that the Vámos matroid is non-algebraic.

On the chromatic number of regular matroids

Abstract We extend to all regular matroids the fact that the chromatic number of a graph is k or less, when all vertex degrees are less than k . If there is a covering of a regular matroid by cocircuits of cardinality less than k , then the chromatic number of the matroid is k or less. A similar result holds for the critical exponent of a vector matroid over GF ( q ).

A class of non-algebraic matroids of rank three

We obtain an infinite class of simple non-algebraic matroids of rank 3, the minors of which are vector matroids and therefore algebraic. We prove that the matroids are non-algebraic with the aid of a theory of harmonic conjugates in full algebraic combinatorial geometries [7].

Matroids algebraic over F(t) are algebraic over F

In his thesis [3] M. J. Piff conjectured that a matroid, which is algebraic over a fieldFit) witht transcendent overF, must be algebraic overF. Two proofs have appeared, one by Shameeva [5] and another one by the author [2], but both are unsatisfactory. In this paper I will settle conjecture by applying a theorem of Seidenberg.

On balance in group graphs

We propose a generalization of signed graphs, called group graphs. These are graphs regarded as symmetric digraphs with a group element s(u, v) called the signing associated with each arc (u, v) such that s(u, v) s (v, u) = 1. A group graph is balanced if the product s(v1, v2) s (v2, v3) …s(um, v1) = 1 for each cycle v1, v2,…, vm, v1 in the graph. Let G denote the graph, F the group (not necessarily commutative), and s the signing. Then the group graph is denoted by (G, F, s). Given a group graph (G, F, s), which need not be balanced, we define the deletion index δ(G, F, s) as the minimum cardinality of a deletion set, which is a subset of lines whose deletion results in a balanced group graph. Similarly we define the alteration index γ(G, F, s) as the minimum cardinality of a alteration set, which is a set of lines {u, v} in the graph the values s(u, v) and s(v, u) of which can be changed so that the new group graph is balanced. When F is the group of order 2, we obtain a signed graph. Generalizing results known for signed graphs, we prove that minimal deletion sets are minimal change sets, which implies the equality δ(G, F, s) = γ(G, F, s) for all G, F, and s. We also prove the in-equality δ (G, F, s) ≤ (n - 1)q/n, where q is the number of lines in the graph G and n is the order of the group F. We conclude by studying δ for signed complete bigraphs Kn,n when the signing is determined from a Hadamard matrix.

On harmonic conjugates in full algebraic combinatorial geometries

Abstract The points of a dense algebraic combinatorial geometry are equivalence classes of transcendentals over a field F in the algebraic closure of a transcendental extension of F . Two transcendentals represent the same point when they are algebraically dependent over F . If x and y are two algebraically independent transcendentals over F the points of the algebraic closure of the field F ( x , y ) belong to a line. Planes are defined similarly. By analogy with classical projective geometry, we define harmonic conjugates with respect to 2 points on a line. We prove the existence and uniqueness of the harmonic conjugate of a point with respect to two other points on a line. The main tool is a lemma by Ingleton and Main in [3].