The codegree density γ(F ) of an r-graph F is the largest number γ such that there are F -free r-graphs G on n vertices such that every set of r−1 vertices is contained in at least (γ−o(1))n edges. When F = PG2(2) is the Fano plane Mubayi showed that γ(F ) = 1/2. This paper studies γ(PGm(q)) for further values of m and q. In particular we have an upper bound γ(PGm(q)) ≤ 1− 1/m for any projectiv...

A Fano colouring is a colouring of the edges of a cubic graph by points of the Fano plane such that the colours of any three mutually adjacent edges form a line of the Fano plane. It has recently been shown by Holroyd and Škoviera (J. Combin. Theory Ser. B, to appear) that a cubic graph has a Fano colouring if and only if it is bridgeless. In this paper we prove that six, and conjecture that fo...

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let (e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, A g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K 3,3' 0 1984 Academic Press, Inc.

Moreover, the only extremal configuration can be obtained by partitioning an n-element set into two almost equal parts, and taking all the triples that intersect both of them. This extends an earlier result of de Caen and Füredi, and proves an old conjecture of V. Sós. In addition, we also prove a stability result for the Fano plane, which says that a 3-uniform hypergraph with density close to ...

The Octonions and their multiplication table, the Fano Plane, bear an isomorphic relationship to the Pythagorean music system, which this paper shows is not a coincidence. In fact, the Octonions vibrate at one stage during the formation of new particles, in what might be termed the Octonion Song. This explains why Octonion structure matches musical structure. This paper explains how this happen...

Given a 3-graph H , let ex2(n,H) denote the maximum value of the minimum co-degree of a 3-graph on n vertices which does not contain a copy of H . Let F denote the Fano plane, which is the 3-graph {axx, ayy, azz , xyz , xyz, xyz, xyz }. Mubayi (2005) [14] proved that ex2(n, F) = (1/2 + o(1))n and conjectured that ex2(n, F) = ⌊n/2⌋ for sufficiently large n. Using a very sophisticated quasirandom...

Given a seven-element set X = {1, 2, 3, 4, 5, 6, 7}, there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from X—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s, 0 ≤...

Finite projective geometries, especially the Fano plane, have been observed to arise in the context of certain quantum gate operators. We use Clifford algebras to explain why these geometries, both planar and higher dimensional, appear in the context of multi-qubit composite systems.

This correspondence studies an estimator of the conditional support of a distribution underlying a set of i.i.d. observations. The relation with mutual information is shown via an extension of Fano’s theorem in combination with a generalization bound based on a compression argument. Extensions to estimating the conditional quantile interval, and statistical guarantees on the minimal convex hull...

Since the seminal paper of Kötter and Kschischang [20] there is a still growing interest in subspace codes, which are sets of subspaces of the Fq-vector space F n q . If all subspaces, which play the role of the codewords, have the same dimension, say k, then one speaks of constant dimension codes. The most commonly used distance measures for subspace codes, motivated by an information-theoreti...