On the upper bound of the size of the r-cover-free families

Let T (r; n) denote the maximum number of subsets of an n-set satisfying the condition in the title. It is proved in a purely combinatorial way, that for n sufficiently large log 2 T (r; n) n 8 \Delta log 2 r r 2 holds. 1. Introduction The notion of the r-cover-free families was introduced by Kautz and Singleton in 1964 [17]. They initiated investigating binary codes with the property that the disjunction of any r (r 2) codewords are distinct (UD r codes). This led them to studying the binary codes with the property that none of the codewords is covered by the disjunction of r others (Superimposed codes, ZFD r codes; P. Erdos, P. Frankl and Z. Furedi called the correspondig set system r-cover-free in [7]). Since that many results have been proved about the maximum size of these codes. Various authors studied these problems basically from three different points of view, and these three lines of investigations were almost independent of each other. This is why many results were ..

On the dd-cluster generalization of Erd\H{o}s-Ko-Rado

If $2 \le d \le k$ and $n \ge dk/(d-1)$, a $d$-cluster is defined to be a collection of $d$ elements of ${[n] \choose k}$ with empty intersection and union of size no more than $2k$. Mubayi conjectured that the largest size of a $d$-cluster-free family $\mathcal{F} \subset {[n] \choose k}$ is ${n-1 \choose k-1}$, with equality holding only for a maximum-sized star. Here we prove two results. The first resolves Mubayi's conjecture and proves a slightly stronger result, thus completing a new generalization of the Erd\H{o}s-Ko-Rado Theorem. The second shows, by a different technique, that for a slightly more limited set of parameters only a very specific kind of $d$-cluster need be forbidden to achieve the same bound

New results on simplex-clusters in set systems

A $d$-simplex is defined to be a collection $A_1,\dots,A_{d+1}$ of subsets of size $k$ of $[n]$ such that the intersection of all of them is empty, but the intersection of any $d$ of them is non-empty. Furthemore, a $d$-cluster is a collection of $d+1$ such sets with empty intersection and union of size $\le 2k$, and a $d$-simplex-cluster is such a collection that is both a $d$-simplex and a $d$-cluster. The Erd\H{o}s-Chv\'{a}tal $d$-simplex Conjecture from 1974 states that any family of $k$-subsets of $[n]$ containing no $d$-simplex must be of size no greater than ${n -1 \choose k-1}$. In 2011, Keevash and Mubayi extended this conjecture by hypothesizing that the same bound would hold for families containing no $d$-simplex-cluster. In this paper, we resolve Keevash and Mubayi's conjecture for all $4 \le d+1 \le k \le n/2$, which in turn resolves all remaining cases of the Erd\H{o}s-Chv\'{a}tal Conjecture except when $n$ is very small (i.e. $n < 2k$).Comment: 8 page

On Mubayi's Conjecture and conditionally intersecting sets

Mubayi's Conjecture states that if $\mathcal{F}$ is a family of $k$-sized subsets of $[n] = \{1,\ldots,n\}$ which, for $k \geq d \geq 2$, satisfies $A_1 \cap\cdots\cap A_d \neq \emptyset$ whenever $|A_1 \cup\cdots\cup A_d| \leq 2k$ for all distinct sets $A_1,\ldots,A_d \in\mathcal{F}$, then $|\mathcal{F}|\leq \binom{n-1}{k-1}$, with equality occurring only if $\mathcal{F}$ is the family of all $k$-sized subsets containing some fixed element. This paper proves that Mubayi's Conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's Conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between $(i,j)$-unstable families and $(j,i)$-unstable families. Generalising previous intersecting conditions, we introduce the $(d,s,t)$-conditionally intersecting condition for families of sets and prove general results thereon. We conjecture on the size and extremal structures of families $\mathcal{F}\in\binom{[n]}{k}$ that are $(d,2k)$-conditionally intersecting but which are not intersecting, and prove results related to this conjecture. We prove fundamental theorems on two $(d,s)$-conditionally intersecting families that generalise previous intersecting families, and we pose an extension of a previous conjecture by Frankl and F\"uredi on $(3,2k-1)$-conditionally intersecting families. Finally, we generalise a classical result by Erd\H{o}s, Ko and Rado by proving tight upper bounds on the size of $(2,s)$-conditionally intersecting families $\mathcal{F}\subseteq 2^{[n]}$ and by characterising the families that attain these bounds. We extend this theorem for certain parametres as well as for sufficiently large families with respect to $(2,s)$-conditionally intersecting families $\mathcal{F}\subseteq 2^{[n]}$ whose members have at most a fixed number $u$ members

Shadows and intersections in vector spaces

AbstractWe prove a vector space analog of a version of the Kruskal–Katona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erdős–Ko–Rado theorem for vector spaces