Cross-intersecting sub-families of hereditary families

Families ${\mathcal{A}}_1,{\mathcal{A}}_2,\mathrm{.}\mathrm{.}\mathrm{.},{\mathcal{A}}_k$ of sets are said to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},k\}$ with $i\mathrm{\ne }j$, any set in ${\mathcal{A}}_i$ intersects any set in ${\mathcal{A}}_j$. For a finite set $X$, let $2^X$ denote the \emph{power set of $X$} (the family of all subsets of $X$). A family $\mathcal{H}$ is said to be \emph{hereditary} if all subsets of any set in $\mathcal{H}$ are in $\mathcal{H}$; so $\mathcal{H}$ is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family $\mathcal{H}\mathrm{\ne }\{\mathrm{\varnothing }\}$ of $2^X$ and any $k\ge \mathrm{\mid }X\mathrm{\mid }+1$, both the sum and product of sizes of $k$ cross-intersecting sub-families $$\mathcal{A}$_1, $\mathcal{A}$_2, …, $\mathcal{A}$_k$(notnecessarilydistinctornon-empty)of$$\mathcal{H}$$aremaximaif$$\mathcal{A}$_1 = $\mathcal{A}$_2 = … = $\mathcal{A}$_k = $\mathcal{S}$for some largest \emph{star$\mathcal{S}$$of$$\mathcal{H}$} (a sub-family of$\mathcal{H}$$ whose sets have a common element). We prove this for the case when $\mathcal{H}$ is \emph{compressed with respect to an element $x$ of $X$}, and for this purpose we establish new properties of the usual \emph{compression operation}. For the product, we actually conjecture that the configuration $$\mathcal{A}$_1 = $\mathcal{A}$_2 = … = $\mathcal{A}$_k = $\mathcal{S}$$isoptimalforanyhereditary$$\mathcal{H}$$ and any $k\ge 2$, and we prove this for a special case too.