A triple system is cancellative if no three of its distinct edges satisfy A ∪ B = A ∪ C. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is cancellative. We prove that almost all cancellative triple systems with vertex set [n] are tripartite. This sharpens a theorem of N...

Let Jt be the class of binary matroids without a Fano plane as a submatroid. We show that every supersolvable matroid in JÍ is graphic, corresponding to a chordal graph. Then we characterize the case that the modular join of two matroids is supersolvable. This is used to study modular flats and modular joins of binary supersolvable matroids. We decompose supersolvable matroids in JH as modular ...

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n+4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of siz...

An Sq[t,k,v] q-Steiner system is a collection of k-dimensional subspaces of the v-dimensional vector space Fq over the finite field Fq with q elements, called blocks, such that each t-dimensional subspace of Fq is contained in exactly one block. The smallest admissible parameters for which a q-Steiner system could exist is S2[2,3,7]. Up to now the issue whether q-Steiner systems with these para...

A conjecture of V. Sós [3] is proved that any set of 34 (n 3 ) + cn2 triples from an n-set, where c is a suitable absolute constant, must contain a copy of the Fano configuration (the projective plane of order two). This is an asymptotically sharp estimate. Given a 3-uniform hypergraph F , let ex3(n,F) denote the maximum possible size of a 3-uniform hypergraph of order n that does not contain a...

Let F 7 denote the Fano matroid and M be a simple connected binary matroid such that every cocircuit of M has size at least d 3. We show that if M does not have an F 7-minor, M 6 = F 7 , and d = 2 f5; 6; 7; 8g, then M has a circuit of size at least minfr(M) + 1; 2dg. We conjecture that the latter result holds for all d 3.

We present orientation dependence and tuning of Fano resonances in plasmonic gold nanorod heptamer nanostructures. The heptamer structures were formed by surrounding a central circular nanoparticle by six satellite nanorods in a hexagonal arrangement and the heptamer structures were then embedded in a thin flexible membrane. The Fano resonance was found to depend on the orientation of the nanor...

T. O. Wehling, ∗ H. P. Dahal, A. I. Lichtenstein, M. I. Katsnelson, H. Manoharan, and A. V. Balatsky 5, † 1. Institut für Theoretische Physik I, Universität Hamburg, D-20355 Hamburg, Germany Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545,USA Institute for Molecules and Materials, Radboud University Nijmegen, NL-6525 AJ Nijmegen, The Netherlands 4 Department o...

288 triangles are strictly biscribed by a plane quartic curve C ⊂ P = P(C). Two computations of this number will be presented. This number 288 = 36× 8 is related with an even theta characteristic of C and with a Fano threefold V22 of genus twelve. In fact there is a natural birational correspondence between the moduli of V22’s and that of plane quartics. This correspondence led the author to a ...

We give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces. The Fano surface of the Fermat cubic of P is the only one to contain 30 elliptic curves...