On the Size of K-Cross-Free Families

Two subsets A,B of an n-element ground set X are said to be crossing, if none of the four sets A∩B, A\B, B\A and X\(A∪B) are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of X does not contain k pairwise crossing elements, then |F|=O(kn). For k=2 and 3, the conjecture is true, but for larger values of k the best known upper bound, due to Lomonosov, is |F|=O(knlogn). In this paper, we improve this bound for large n by showing that |F|=Ok(nlog*n) holds, where log* denotes the iterated logarithm function. © 2018 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Natur

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

Families of sequences with good family complexity and cross-correlation measure

In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.Comment: 13 pages. Comments are welcome