Intersection properties of subsets of integers

Let {A1,...,AN} be a family of subsets of {1, 2,...,n}. For a fixed integer k we assume that keyable*** is an arithmetic progression of ⩾k elements for every 1 ⩽ i < j ⩽ N. We would like to determine the maximum of N. For k = 0, R. L. Graham and the authors have proved that N=⩽(n3)+(n2)+(n1)+1For k ⩾ 2, the extremal and asymptotically extremal systems have (π224+o(1))n2setsFor k = 1, the maximum is between (n2)+1and(π224+12o(1))n2We conjecture that the lower bound is sharp

Intersection properties of subsets of integers

Abstract Let N k be the maximal integer such that there exist subsets A 1 , . . . , A N k ⊆ {1, 2, . . . , n} for which A i ∩ A j is an arithmetic progression of length at least k for every 1 ≤ i &lt; j ≤ N k . R. L. Graham, M. Simonovits and V. T. Sós gave the exact value of N 0 . For k ≥ 2, Simonovits and T. Sós determined the asymptotic behavior of N k . In this paper we prove a conjecture of Simonovits and T. Sós concerning the asymptotic value of N 1 . We show that Moreover, we slightly improve the best known construction, thus disproving their conjecture on the exact extremal system

A note on the intersection properties of subsets of integers

From the introduction: "According to the de Bruijn-Erdős theorem, if A1,⋯,AN are subsets of an n-element set S and |Ai∩Aj|=1 for i≠j (where |X| denotes the cardinality of X), then N≤n. This result is sharp, e.g., if S={1,⋯,n}=[1,n] and A1={1,n}, A2={2,n},⋯,An−1={n−1,n}, and An={1,2,⋯,n−1}, then Ai∩Aj=1 for 1≤i<j≤n. Many similar theorems have been proved for sets. One could also ask what analogous results can be proved if the Ai have some extra structure and the condition on the intersection also refers to this structure. For example, it has been proved [Simonovits and Sós, Problèmes combinatoires et théorie des graphes (Orsay, 1976), pp. 389–391, CNRS, Paris, 1980: MR0540021 (80i:05062a)], that if A1,⋯,AN are graphs on the same n vertices and the intersection of two graphs Ai and Aj is defined as the graph without isolated vertices whose edges are the common edges of Ai and Aj, then the condition `Ai∩Aj is a (nonempty) cycle for 1≤i<j≤N' implies that N≤(n2)−2, which is again sharp. Here we investigate the case in which A1,⋯,AN is a system of subsets of {1,⋯,n} and the intersection condition is of a number-theoretic type.'

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

Counting packings of generic subsets in finite groups

A packing of subsets $\mathcal S_1,..., \mathcal S_n$ in a group $G$ is a sequence $(g_1,...,g_n)$ such that $g_1\mathcal S_1,...,g_n\mathcal S_n$ are disjoint subsets of $G$. We give a formula for the number of packings if the group $G$ is finite and if the subsets $\mathcal S_1,...,\mathcal S_n$ satisfy a genericity condition. This formula can be seen as a generalization of the falling factorials which encode the number of packings in the case where all the sets $\mathcal S_i$ are singletons

Large Sets in Boolean and Non-Boolean Groups and Topology

Right and left thick, syndetic, piecewise syndetic, and fat sets in groups are studied. The main concern is the interplay between such sets in Boolean groups. Natural topologies closely related to fat sets are also considered, which leads to interesting relations between fat sets and ultrafilters

The Isomorphism Relation Between Tree-Automatic Structures

An $\omega$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $\omega$-tree-automatic structures. We prove first that the isomorphism relation for $\omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for $\omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is neither a $\Sigma_2^1$-set nor a $\Pi_2^1$-set

On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure