Peter Frankl

Families of finite sets in which no set is covered by the union ofr others

AbstractLetfr(n, k) denote the maximum number ofk-subsets of ann-set satisfying the condition in the title. It is proved that

The Johnson-Lindenstrauss Lemma and the sphericity of some graphs

f_1 (n,r(t - 1) + 1 + d)\underset{\raise0.3em\hbox{

On the maximum number of permutations with given maximal or minimal distance

\smash{\scriptscriptstyle-}

The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent

}}{ \leqslant } (_{ t}^{n - d} )/(_{ t}^{k - d} )

Families of Finite Sets in Which No Set Is Covered by the Union of Two Others

wheneverd=0, 1 ord≦r/2t2 with equality holding iff there exists a Steiner systemS(t, r(t−1)+1,n−d). The determination offr(n, 2r) led us to a new generalization of BIBD (Definition 2.4). Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family.

Geometrical realization of set systems and probabilistic communication complexity

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ0, μ1, ..., μs are distinct residues modp (p is a prime) such thatk ≡ μ0 (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μi (modp) for somei, 1 ≦i≦s, then |ℱ|≦(sn).As a consequence we show that ifRn is covered bym sets withm<(1+o(1)) (1.2)n then there is one set within which all the distances are realised.It is left open whether the same conclusion holds for compositep.

Hypergraphs do not jump

A simple short proof of the Johnson-Lindenstrauss lemma (concerning nearly isometric embeddings of finite point sets in lower-dimensional spaces) is given. This result is applied to show that if G is a graph on n vertices and with smallest eigenvalue i then its sphericity sph(G) is less than cA2 log n. It is also proved that if G or its complement is a forest then sph(G) < c log n holds. Q 19%8 Academic Press, Inc.

The Erdo¨s-Ko-Rado theorem for vector spaces

Let us denote by R(k, ⩾ λ)[R(k, ⩽ λ)] the maximal number M such that there exist M different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ⩽ λ) ⩽ kR(k − 1, ⩽ λ − 1), R(k, ⩾ λ) ⩾ R(k, ⩽ λ − 1) ⩽ k!, R(k, ⩾ λ) ⩽ kR(k − 1, ⩾ λ − 1). We show: R(k, ⩾ k − 2) = 2, R(k, ⩾ 1) = (k − 1)!, R(pm, ⩾ 2) = (pm − 2)!, R(pm + 1, ⩾ 3) = (pm − 2)!, R(k, ⩽ k − 3) = k!2, R(k, ⩽ 0) = k, R(pm, ⩽ 1) = pm(pm − 1), R(pm + 1, ⩽ 2) = (pm + 1)pm(pm − 1). The exact value of R(k, ⩾ λ) is determined whenever k ⩾ k0(k − λ); we conjecture that R(k, ⩾ λ) = (k − λ)! for k ⩾ k0(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ⩾ λ) − R(k, ⩽ λ)| is attained for λ = (k2) + O(klog k).