Cycles in Oriented 3-graphs

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\frac{n^2}{3}(1+o(1))$, in complete contrast to the case of 2-tournaments.Comment: 12 page

Improved Bounds for the Graham-Pollak Problem for Hypergraphs

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$, and we write $c_r$ for the least number such that $f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$. It was known that $c_r < 1$ for each even $r \geq 4$, but this was not known for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our method also shows that $c_r \rightarrow 0$, answering another open problem

A Note on Hamiltonian-Intersecting Families of Graphs

How many graphs on an $n$-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly $1/2^{n-1}$ of all graphs. Our aim in this short note is to give a 'directed' version of this result; we show that a family of oriented graphs such that any two have strongly-connected intersection has size at most $1/3^n$ of all oriented graphs. We also show that a family of graphs such that any two have Hamiltonian intersection has size at most $1/2^n$ of all graphs, verifying a conjecture of the above authors.Comment: 5 page

A collection of open problems in celebration of Imre Leader's 60th birthday

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics

Robust and Task-Independent Spatial Profile of the Visual Word Form Activation in Fusiform Cortex

Written language represents a special category of visual information. There is strong evidence for the existence of a cortical region in ventral occipitotemporal cortex for processing the visual form of written words. However, due to inconsistent findings obtained with different tasks, the level of specialization and selectivity of this so called visual word form area (VWFA) remains debated. In this study, we examined category selectivity for Chinese characters, a non-alphabetic script, in native Chinese readers. In contrast to traditional approaches of examining response levels in a restricted predefined region of interest (ROI), a detailed distribution of the BOLD signal across the mid-fusiform cortical surface and the spatial patterns of responses to Chinese characters were obtained. Results show that a region tuned for Chinese characters could be consistently found in the lateral part of the left fusiform gyrus in Chinese readers, and this spatial pattern of selectivity for written words was not influenced by top-down tasks such as phonological or semantic modulations. These results provide strong support for the robust spatial coding of category selective response in the mid-fusiform cortex, and demonstrate the utility of the spatial distribution analysis as a more meaningful approach to examine functional magnetic resonance imaging (fMRI) data

Genome-wide regulation of innate immunity by juvenile hormone and 20-hydroxyecdysone in the Bombyx fat body

<p>Abstract</p> <p>Background</p> <p>Insect innate immunity can be affected by juvenile hormone (JH) and 20-hydroxyecdysone (20E), but how innate immunity is developmentally regulated by these two hormones in insects has not yet been elucidated. In the silkworm, <it>Bombyx mori</it>, JH and 20E levels are high during the final larval molt (4 M) but absent during the feeding stage of 5^thinstar (5 F), while JH level is low and 20E level is high during the prepupal stage (PP). Fat body produces humoral response molecules and hence is considered as the major organ involved in innate immunity.</p> <p>Results</p> <p>A genome-wide microarray analysis of <it>Bombyx </it>fat body isolated from 4 M, 5 F and PP uncovered a large number of differentially-expressed genes. Most notably, 6 antimicrobial peptide (AMP) genes were up-regulated at 4 M versus PP suggesting that <it>Bombyx </it>innate immunity is developmentally regulated by the two hormones. First, JH treatment dramatically increased AMP mRNA levels and activities. Furthermore, 20E treatment exhibited inhibitory effects on AMP mRNA levels and activities, and RNA interference of the 20E receptor <it>EcR</it>-<it>USP </it>had the opposite effects to 20E treatment.</p> <p>Conclusion</p> <p>Taken together, we demonstrate that JH acts as an immune-activator while 20E inhibits innate immunity in the fat body during <it>Bombyx </it>postembryonic development.</p