On a covering problem in the hypercube

In this paper, we address a particular variation of the Tur\'an problem for the hypercube. Alon, Krech and Szab\'o (2007) asked "In an n-dimensional hypercube, Qn, and for l < d < n, what is the size of a smallest set, S, of Q_l's so that every Q_d contains at least one member of S?" Likewise, they asked a similar Ramsey type question: "What is the largest number of colors that we can use to color the copies of Q_l in Q_n such that each Q_d contains a Q_l of each color?" We give upper and lower bounds for each of these questions and provide constructions of the set S above for some specific cases.Comment: 8 page

On the decomposition of random hypergraphs

For an $r$-uniform hypergraph $H$, let $f(H)$ be the minimum number of complete $r$-partite $r$-uniform subhypergraphs of $H$ whose edge sets partition the edge set of $H$. For a graph $G$, $f(G)$ is the bipartition number of $G$ which was introduced by Graham and Pollak in 1971. In 1988, Erd\H{o}s conjectured that if $G \in G(n,1/2)$, then with high probability $f(G)=n-\alpha(G)$, where $\alpha(G)$ is the independence number of $G$. This conjecture and related problems have received a lot of attention recently. In this paper, we study the value of $f(H)$ for a typical $r$-uniform hypergraph $H$. More precisely, we prove that if $(\log n)^{2.001}/n \leq p \leq 1/2$ and $H \in H^{(r)}(n,p)$, then with high probability $f(H)=(1-\pi(K^{(r-1)}_r)+o(1))\binom{n}{r-1}$, where $\pi(K^{(r-1)}_r)$ is the Tur\'an density of $K^{(r-1)}_r$.Comment: corrected few typos. updated the referenc