Some New Bounds For Cover-Free Families Through Biclique Cover

An $(r,w;d)$ cover-free family $(CFF)$ is a family of subsets of a finite set such that the intersection of any $r$ members of the family contains at least $d$ elements that are not in the union of any other $w$ members. The minimum number of elements for which there exists an $(r,w;d)-CFF$ with $t$ blocks is denoted by $N((r,w;d),t)$. In this paper, we show that the value of $N((r,w;d),t)$ is equal to the $d$-biclique covering number of the bipartite graph $I_t(r,w)$ whose vertices are all $w$- and $r$-subsets of a $t$-element set, where a $w$-subset is adjacent to an $r$-subset if their intersection is empty. Next, we introduce some new bounds for $N((r,w;d),t)$. For instance, we show that for $r\geq w$ and $r\geq 2$ $$ N((r,w;1),t) \geq c{{r+w\choose w+1}+{r+w-1 \choose w+1}+ 3 {r+w-4 \choose w-2} \over \log r} \log (t-w+1),$$ where $c$ is a constant satisfies the well-known bound $N((r,1;1),t)\geq c\frac{r^2}{\log r}\log t$. Also, we determine the exact value of $N((r,w;d),t)$ for some values of $d$. Finally, we show that $N((1,1;d),4d-1)=4d-1$ whenever there exists a Hadamard matrix of order 4d.

💡 Research Summary

The paper establishes a precise equivalence between the minimum size of an ((r,w;d)) cover‑free family (CFF), denoted (N((r,w;d),t)), and the (d)-biclique covering number of a bipartite graph (I_t(r,w)). In (I_t(r,w)) the left part consists of all (w)-subsets of a (t)-element ground set, the right part of all (r)-subsets, and an edge joins a pair whenever the subsets are disjoint. By showing that covering every edge of this graph at least (d) times with complete bipartite subgraphs (bicliques) requires exactly the same number of vertices as constructing a CFF with (t) blocks, the authors translate a combinatorial design problem into a graph‑theoretic covering problem.

Using this translation they derive a new lower bound for the case (d=1) and (r\ge w), (r\ge2): \