Packing k-partite k-uniform hypergraphs

Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. For any fixed $H$ let $\delta(H, n)$ be the minimum $\delta$ such that any $k$-graph $G$ on $n$ vertices with minimum codegree $\delta(G) \geq \delta$ contains a perfect $H$-packing. The problem of determining $\delta(H, n)$ has been widely studied for graphs (i.e. $2$-graphs), but little is known for $k \geq 3$. Here we determine the asymptotic value of $\delta(H, n)$ for all complete $k$-partite $k$-graphs $H$, as well as a wide class of other $k$-partite $k$-graphs. In particular, these results provide an asymptotic solution to a question of R\"odl and Ruci\'nski on the value of $\delta(H, n)$ when $H$ is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an $H$-packing covering all but a constant number of vertices of $G$ for any complete $k$-partite $k$-graph $H$.Comment: v2: Updated with minor corrections. Accepted for publication in Journal of Combinatorial Theory, Series

Assessment of the control measures of the category A diseases of Animal Health Law: peste des petits ruminants

EFSA received a mandate from the European Commission to assess the effectiveness of some of the control measures against diseases included in the Category A list according to Regulation (EU) 2016/429 on transmissible animal diseases (‘Animal Health Law’). This opinion belongs to a series of opinions where these control measures will be assessed, with this opinion covering the assessment of control measures for peste des petits ruminants (PPR). In this opinion, EFSA and the AHAW Panel of experts review the effectiveness of: (i) clinical and laboratory sampling procedures, (ii) monitoring period and (iii) the minimum radii of the protection and surveillance zones, and the minimum length of time the measures should be applied in these zones. The general methodology used for this series of opinions has been published elsewhere; nonetheless, the transmission kernels used for the assessment of the minimum radii of the protection and surveillance zones are shown. Several scenarios for which these control measures had to be assessed were designed and agreed prior to the start of the assessment. The monitoring period of 21 days was assessed as effective, except for the first affected establishments detected, where 33 days is recommended. It was concluded that beyond the protection (3 km) and the surveillance zones (10 km) only 9.6% (95% CI: 3.1–25.8%) and 2.3% (95% CI: 1–5.5%) of the infections from an affected establishment may occur, respectively. This may be considered sufficient to contain the disease spread (95% probability of containing transmission corresponds to 5.3 km). Recommendations provided for each of the scenarios assessed aim to support the European Commission in the drafting of further pieces of legislation, as well as for plausible ad-hoc requests in relation to PPR