A note on tilted Sperner families with patterns

Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\mathcal{F} \subset \mathcal{P}([n])$ a \textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \in \mathcal{F}$ with: $$(i) \ \ p|F \setminus G|=q|G \setminus F|, \ \textrm{and}$$ $$(ii) \ f > g \ \textrm{for all} \ f \in F \setminus G \ \textrm{and} \ g \in G \setminus F.$$ Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\sqrt{\log n}}\ \frac{2^n}{\sqrt{n}}).$$ We improve and generalize this result, and prove that the cardinality of every ($p,q$)-tilted Sperner family with patterns on [$n$] is $$O(\sqrt{\log n} \ \frac{2^n}{\sqrt{n}}).$$Comment: 8 page

On Tur\'an problems with bounded matching number

Very recently, Alon and Frankl initiated the study of the maximum number of edges in $n$-vertex $F$-free graphs with matching number at most $s$. For fixed $F$ and $s$, we determine this number apart from a constant additive term. We also obtain several exact results

Rainbow copies of FF in families of HH

We study the following problem. How many distinct copies of $H$ can an $n$-vertex graph $G$ have, if $G$ does not contain a rainbow $F$, that is, a copy of $F$ where each edge is contained in a different copy of $H$? The case $H=K_r$ is equivalent to the Tur\'an problem for Berge hypergraphs, which has attracted several researchers recently. We also explore the connection of our problem to the so-called generalized Tur\'an problems. We obtain several exact results. In the particularly interesting symmetric case where $H=F$, we completely solve the case $F$ is the 3-edge path, and asymptitically solve the case $F$ is a book graph