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Hypergraphs
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RICHARD ANSTEE, UBC Vancouver
Forbidden Berge hypergraphs  [PDF]

Joint work with Santiago Salazar. A \emph{simple} matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a \emph{Berge hypergraph} if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $||{A}||$ denote the number of columns in $A$, we define the extremal function $Bh(m,{ F})=\max\{||{A}||\,:\, A \hbox{ is an }m\hbox{-rowed simple matrix with no Berge hypergraph }F\}$. We determine the asymptotics of $Bh(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determining the maximum number of copies of $K_r$ in a $m$-vertex graph that has no $K_{s,t}$ subgraph, a problem studied by Alon and Shikhelman.

AMIN BAHMANIAN, Illinois State University
On The Existence of Generalized Designs  [PDF]

A set $S$ of $q$-subsets of an $n$-set $X$ is a design with parameters $(n,q,r,\lambda)$ if every $r$-subset of $X$ belongs to exactly $\lambda$ elements of $S$. In other words, a design with parameters $(n,q,r,\lambda)$ is an $n$-vertex $q$-uniform hypergraph in which every $r$-subset of the vertex set belongs to exactly $\lambda$ edges. The existence of a design with parameters $(n,q,r,\lambda)$ is equivalent to a $K_q^r$-decomposition of $\lambda K_n^r$ (the complete $\lambda$-fold $r$-uniform hypergraph of order $n$). By Keevash's Theorem (2014), $\lambda K_n^r$ can be decomposed into $K_q^r$ when some obvious divisibility conditions are satisfied and $n$ is sufficiently large. In this talk, I will discuss a multipartite" version of Keevash's Theorem.

Keywords: hypergraphs, designs, generalized designs, multipartite, amalgamation, detachment

ANDRZEJ CZYGRINOW, Arizona State University
Tight minimum degree condition for tiling a 3-graph with loose cycles  [PDF]

Let $C_t$ denote the loose cycle on $t$ vertices, the 3-uniform hypergraph obtained from a graph cycle $C$ on $t/2$ vertices by inserting a new vertex $v_e$ for every edge $e\in C$. For a 3-uniform hypergraph $H$ let $\delta(H):=\min_{v\in V(H)} |N(v)|$ denote the minimum degree of $H$. We will give a tight condition for $\delta(H)$ which guarantees that a large enough 3-uniform hypergraph $H$ on $n\in tZ$ vertices has $n/t$ vertex disjoint copies of $C_t$. This is a joint work with R. Oursler.