CanaDAM 2017
Université Ryerson, 12 - 16 juin 2017


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.


Atlantic Association for Research in the Mathematical Sciences Centre de recherches mathmatiques The Fields Institute Pacific Institute for the Mathematical Sciences Socit mathmatique du Canada Université Ryerson Office of Naval Research Science and Technology