It is shown that Rota’s basis conjecture follows from a similar conjecture that involves only three bases instead of n bases. Two counterexamples to the analogous conjecture involving only two bases are presented. [Note added 18 May 2005: Colin McDiarmid found a counterxample to Conjecture 2 after reading a previous version of this paper. See the end of the paper for details.]

Let P be a set of n points in R. It was conjectured by Schur that the maximum number of (d− 1)-dimensional regular simplices of edge length diam(P ), whose every vertex belongs to P , is n. We prove this statement under the condition that any two of the simplices share at least d− 2 vertices and we conjecture that this condition is always satisfied.

We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen’s generalized lower bounds, and of Barnette’s lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variab...

We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes’ Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations.

We introduce 3N × 3N Lax pair with spectral parameter for Calogero-Inozemtsev model. The one degree of freedom case appears to have 2 × 2 Lax representation. We derive it from the elliptic Gaudin model via some reduction procedure and prove algebraic integrability. This Lax pair provides elliptic linear problem for the Painlevé VI equation in elliptic form.

We prove bispectral duality for the generalized Calogero–Moser–Sutherland systems related to configurations An,2(m), Cn(l,m). The trigonometric axiomatics of Baker–Akhiezer function is modified, the dual difference operators of rational Macdonald type and the Baker–Akhiezer functions related to both series are explicitly constructed.

We consider the physical properties of elementary excitations of the CalogeroSutherland (CS) model with SU(K) internal symmetry. From the results on the thermodynamics of this model, we obtain the charge, spin, and statistics of elementary excitations. Combining this knowledge and the known results on the dynamics in the spinless CS model, we propose the expression for the dynamical correlation...

I n 1970, Gian-Carlo Rota posed a conjecture predicting a beautiful combinatorial characterization of linear dependence in vector spaces over any given finite field. We have recently completed a fifteen-year research program that culminated in a solution of Rota’s Conjecture. In this article we discuss the conjecture and give an overview of the proof. Matroids are a combinatorial abstraction of...

The Erdős-Faber-Lovász conjecture is the statement that every graph that is the union of n cliques of size n intersecting pairwise in at most one vertex has chromatic number n. Kahn and Seymour proved a fractional version of this conjecture, where the chromatic number is replaced by the fractional chromatic number. In this note we investigate similar fractional relaxations of the Erdős-Faber-Lo...