Some sporadic translation planes of order 11211^2

In \cite{PK}, the authors constructed a translation plane $\Pi$ of order $11^2$ arising from replacement of a sporadic chain $F'$ of reguli in a regular spread $F$ of $PG(3,11)$. They also showed that two more non isomorphic translation planes, called  $\Pi_1$ and $\Pi_{13}$, arise respectively by derivation and double derivation in $F\setminus F'$ which correspond to a further replacement of a regulus with its opposite regulus and a pair of reguli with their opposite reguli, respectively.  In \cite{AL}, the authors proved that the translation complement of $\Pi$ contains a subgroup isomorphic to \SL(2,5). Here, the full collineation group of each of the planes $\Pi$, $\Pi_1$ and $\Pi_{13}$ is determined

On Bruen chains

It is known that a Bruen chain of the three-dimensional projective space $\mathrm{PG}(3,q)$ exists for every odd prime power $q$ at most $37$, except for $q=29$. It was shown by Cardinali et. al (2005) that Bruen chains do not exist for $41\le q\leq 49$. We develop a model, based on finite fields, which allows us to extend this result to $41\leqslant q \leqslant 97$, thereby adding more evidence to the conjecture that Bruen chains do not exist for $q>37$. Furthermore, we show that Bruen chains can be realised precisely as the $(q+1)/2$-cliques of a two related, yet distinct, undirected simple graphs

Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes

We consider topology changing transitions for M-theory compactifications on Calabi-Yau fourfolds with background G-flux. The local geometry of the transition is generically a genus g curve of conifold singularities, which engineers a 3d gauge theory with four supercharges, near the intersection of Coulomb and Higgs branches. We identify a set of canonical, minimal flux quanta which solve the local quantization condition on G for a given geometry, including new solutions in which the flux is neither of horizontal nor vertical type. A local analysis of the flux superpotential shows that the potential has flat directions for a subset of these fluxes and the topologically different phases can be dynamically connected. For special geometries and background configurations, the local transitions extend to extremal transitions between global fourfold compactifications with flux. By a circle decompactification the M-theory analysis identifies consistent flux configurations in four-dimensional F-theory compactifications and flat directions in the deformation space of branes with bundles.Comment: 93 pages; v2: minor changes and references adde

The de Bruijn-Erdős Theorem for Hypergraphs

Abstract Fix integers n ≥ r ≥ 2. A clique partition of is a collection of proper subsets is a partition of . Let cp(n, r) denote the minimum size of a clique partition of . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, We conjecture cp(n, r) = (1 + o(1))n r/2 . This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman-Mullin-Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1 + o(1))n r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n r ) of the r-sets of [n] are covered. We also give an absolute lower bound cp(n, r) when n = q 2 + q + r − 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem