Search

Fix a hypergraph $\mathcal{F}$. A hypergraph $\mathcal{H}$ is called a {\it Berge copy of $\mathcal{F}$} or {\it Berge-$\mathcal{F}$} if we can choose a subset of each hyperedge of $\mathcal{H}$ to obtain a copy of $\mathcal{F}$. A hypergraph $\mathcal{H}$ is {\it Berge-$\mathcal{F}$-free} if it does not contain a subhypergraph which is Berge copy of $\mathcal{F}$. This is a generalization of the usual, graph based Berge hypergraphs, where $\mathcal{F}$ is a graph. In this paper, we study extremal properties of hypergraph based Berge hypergraphs and generalize several results from the graph based setting. In particular, we show that for any $r$-uniform hypregraph $\mathcal{F}$, the sum of the sizes of the hyperedges of a (not necessarily uniform) Berge-$\mathcal{F}$-free hypergraph $\mathcal{H}$ on $n$ vertices is $o(n^r)$ when all the hyperedges of $\mathcal{H}$ are large enough. We also give a connection between hypergraph based Berge hypergraphs and generalized hypergraph Tur\'an problems., Comment: 13 pages

Let F be a graph. We say that a hypergraph H is a Berge- F if there is a bijection f : E ( F ) → E ( H ) such that e ⊆ f ( e ) for every e ∈ E ( F ) . Note that Berge- F actually denotes a class of hypergraphs. The maximum number of edges in an n -vertex r -graph with no subhypergraph isomorphic to any Berge- F is denoted ex r ( n , Berge- F ) . In this paper, we investigate the case when F = K s , t and establish an upper-bound when r ≥ 3 , and a lower-bound when r = 4 and t is large enough compared to s . Additionally, we prove a counting result for r -graphs of girth five that complements the asymptotic formula ex 3 ( n , Berge- { C 2 , C 3 , C 4 } ) = 1 6 n 3 ∕ 2 + o ( n 3 ∕ 2 ) of Lazebnik and Verstraete (2003).

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $\delta_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\H os-Ko-Rado theorem which corresponds to the case $k=1$. Our proof is based on the delta-system method.

Let $L$ be a set of positive integers. We call a (directed) graph $G$ an $L$\emph{-cycle graph} if all cycle lengths in $G$ belong to $L$. Let $c(L,n)$ be the maximum number of cycles possible in an $n$-vertex $L$-cycle graph (we use $\vec{c}(L,n)$ for the number of cycles in directed graphs). In the undirected case we show that for any fixed set $L$, we have $c(L,n)=\Theta_L(n^{\lfloor k/\ell \rfloor})$ where $k$ is the largest element of $L$ and $2\ell$ is the smallest even element of $L$ (if $L$ contains only odd elements, then $c(L,n)=\Theta_L(n)$ holds.) We also give a characterization of $L$-cycle graphs when $L$ is a single element. In the directed case we prove that for any fixed set $L$ we have $\vec{c}(L,n)=(1+o(1))(\frac{n-1}{k-1})^{k-1}$, where $k$ is the largest element of $L$. We determine the exact value of $\vec{c}(\{k\},n)$ for every $k$ and characterize all graphs attaining this maximum.

We examine the following version of a classic combinatorial search problem introduced by Renyi: Given a finite set X of n elements we want to identify an unknown subset Y of X , which is known to have exactly d elements, by means of testing, for as few as possible subsets A of X , whether A intersects Y or not. We are primarily concerned with the non-adaptive model, where the family of test sets is specified in advance, in the case where each test set is of size at most some given natural number k . Our main results are nearly tight bounds on the minimum number of tests necessary when d and k are fixed and n is large enough.

Let $E(G)$ and $V(G)$ denote the edge set and vertex set of a (hyper)graph $G$. Let $G$ be a graph and $\mathcal{H}$ be a hypergraph. We say that a hypergraph $\mathcal{H}$ is a Berge-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subset f(e)$. This generalizes the established definitions of “Berge path” and “Berge cycle” to general graphs. For a fixed graph $G$ we examine the maximum possible size of a hypergraph with no Berge-$G$ as a subhypergraph. In the present paper we prove general bounds for this maximum when $G$ is an arbitrary graph. We also consider the specific case when $G$ is a complete bipartite graph and prove an analogue of the Kovari--Sos--Turan theorem. In case $G$ is $C_4$, we improve the bounds given by Gyori and Lemons [Discrete Math., 312, (2012), pp. 1518--1520].

For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from $F$ by replacing each edge of $F$ with a hyperedge containing it. A hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity $\sum_{h \in E(\mathcal{H})} |h|$. Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\mathcal{H}$ has size at least the Ramsey number of $F$ and at most $o(n)$, the weight of $\mathcal{H}$ is $o(n^2)$. This result is best possible in some sense. Along the way, we study other weight functions, and strengthen results of Gerbner and Palmer; and Gr\'osz, Methuku and Tompkins., Comment: 7 pages. Results are slightly strengthened, and proofs are made simpler

Introduction: The close functional relationship between areas 3b and 1 of the somatosensory cortex is based on their reciprocal connections indicating that tactile sensation depends on the interaction of these two areas. Aim: The aim of the authors was to explore this neuronal circuit at the level of the distal finger pad representation. Method: The study was made by bidirectional tract tracing aided by neurophysiological mapping in squirrel monkeys (Saimiri sciureus). Results: Inter-areal connections between the two areas preferred the homologues representations. However, intra-areal connections were formed between the neighboring finger pad representations supporting the physiological observations. Interestingly, the size of the local input area of the injected cortical micro-region, which differed in the two areas, represented the same skin area. Conclusions: The authors propose that intra-areal connections are important in integrating information across fingers, while inter-areal connections are important in maintaining input localization during hand movement. Orv. Hetil., 2016, 157(33), 1320–1325.

For a graph F , a hypergraph H is a Berge copy of F (or a Berge- F in short), if there is a bijection f : E ( F ) → E ( H ) such that for each e ∈ E ( F ) we have e ⊂ f ( e ) . A hypergraph is Berge- F -free if it does not contain a Berge copy of F . We denote the maximum number of hyperedges in an n -vertex r -uniform Berge- F -free hypergraph by ex r ( n , Berge- F ) . In this paper we prove two general lemmas concerning the maximum size of a Berge- F -free hypergraph and use them to establish new results and improve several old results. In particular, we give bounds on ex r ( n , Berge- F ) when F is a path (reproving a result of Győri, Katona and Lemons), a cycle (extending a result of Furedi and Ozkahya), a theta graph (improving a result of He and Tait), or a K 2 , t (extending a result of Gerbner, Methuku and Vizer). We establish new bounds when F is a clique (which implies extensions of results by Maherani and Shahsiah and by Gyarfas) and when F is a general tree.

The synthesis of 2,5,5-trisubstituted tetrahydrofuran rings was accomplished via the Mukaiyama aerobic oxidative cyclization of tertiary 5-pentenols employing the Co(nmp) 2 catalyst. A variety of THFs were formed in moderate to good yield and diastereoselectivity. The method developed herein was successfully applied to an enantioselective total synthesis of cyclocapitelline.

For positive integers k and d such that 4 ź k < d and k ź 5 , we determine the maximum number of rainbow colored copies of C 4 in a k -edge-coloring of the d -dimensional hypercube Q d . Interestingly, the k -edge-colorings of Q d yielding the maximum number of rainbow copies of C 4 also have the property that every copy of C 4 which is not rainbow is monochromatic.

Functions of the cerebral cortex emerge via interactions of horizontally distributed neuronal populations within and across areas. However, the connectional underpinning of these interactions is not well understood. The present study explores the circuitry of column-size cortical domains within the hierarchically organized somatosensory cortical areas 3b and 1 using tract tracing and optical intrinsic signal imaging (OIS). The anatomical findings reveal that feedforward connections exhibit high topographic specificity, while intrinsic and feedback connections have a more widespread distribution. Both intrinsic and inter-areal connections are topographically oriented across the finger representations. Compared to area 3b, the low clustering of connections and small cortical magnification factor supports that the circuitry of area 1 scaffolds a sparse functional representation that integrates peripheral information from a large area that is fed back to area 3b. Fast information exchange between areas is ensured by thick axons forming a topographically organized, reciprocal pathway. Moreover, the highest density of projecting neurons and groups of axon arborization patches corresponds well with the size and locations of the functional population response reported by OIS. The findings establish connectional motifs at the mesoscopic level that underpin the functional organization of the cerebral cortex.

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Turán number of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.