Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA ₁, ...,A _k are distinct subsets of a set of cardinalityn, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, andk>n, then some two ofA ₁, ...,A _k have empty intersection. We prove a strengthening, that at leastk /n ofA ₁, ...,A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Two-point concentration in random geometric graphs

A random geometric graph G _n is constructed by taking vertices X ₁,…,X _n ∈ℝ ^d at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between X _i and X _j if ‖X _i -X _j ‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r=r(n) is chosen such that nr ^d = o(lnn) then the probability distribution of the clique number ω(G _n ) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number χ(G _n ). The author was partially supported by EPSRC, the Department of Statistics, Bekkerla-Bastide fonds, Dr. Hendrik Muller’s Vaderlandsch fonds, and Prins Bernhard Cultuurfonds.     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Inclusion-exclusion inequalities

Ifμ is a positive measure, andA ₂, ...,A _n are measurable sets, the sequencesS ₀, ...,S _n andP _[0], ...,P _[n] are related by the inclusion-exclusion equalities. Inequalities among theS _i are based on the obviousP _[k]≧0. Letting =the average average measure of the intersection ofk of the setsA _i , it is shown that (−1) ^k Δ ^k M _i ≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS ₀=1, whenS ₁≧N−1, and for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM ₀, ...,M _n of sufficiently large length if and only if for 0<t<1.     

Extreme value limit laws in the nonidentically distributed case

LetX ₁, ...,X _n be independent random variables, letF _i be the distribution function ofX _i (1≦i≦n) and letX _1n ≦... ≦X _nn be the corresponding order statistics. We consider the statisticsX _kn, wherek=k(n),k/n → 1 andn−k → ∞. Under some additional restrictions concerning the behaviour of the sequences {a _n>0,b _n,k(n),F _n} we characterize the class of all distribution functionsH such that Prob{(X _kn −b _n )/a _n <x)}→H. Dedicated to the Memory of N. V. Smirnov (1900–1966)     

Matrices with the edmonds—Johnson property

A matrixA=(a _ij ) has theEdmonds—Johnson property if, for each choice of integral vectorsd ₁,d ₂,b ₁,b ₂, the convex hull of the integral solutions ofd ₁≦x≦d ₂,b ₁≦Ax≦b ₂ is obtained by adding the inequalitiescx≦|δ|, wherec is an integral vector andcx≦δ holds for each solution ofd ₁≦x≦d ₂,b ₁≦Ax≦b ₂. We characterize the Edmonds—Johnson property for integral matricesA which satisfy for each (row index)i. A corollary is that ifG is an undirected graph which does not contain any homeomorph ofK ₄ in which all triangles ofK ₄ have become odd circuits, thenG ist-perfect. This extends results of Boulala, Fonlupt, Sbihi and Uhry. First author’s research supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).     

Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function

LetX ₁,X ₂, ...,X _n be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY ₁,Y ₂, ...,Y _n be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X _i }. We can only observeZ _i =min(X _i ,Y _i ), and . LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival functionS(t) are given. It is shown that (1)P(δ_(n)=1 i.o.)=0 ifF(τ _H ) < 1 andP(δ _n =0 i.o. )=0 ifG(τH) > 1 where δ_(n) is the corresponding indicator function of and have the same order a.s., where {T _n } is a sequence of constants such that 1−H(T _n )=n ^−α(logn)^β(log logn)^γ.     

Lifts, Discrepancy and Nearly Optimal Spectral Gap*

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π :H →G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n “new” eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range (if true, this is tight, e.g. by the Alon–Boppana bound). Here we show that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range for some constant c. This leads to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue . The proof uses the following lemma (Lemma 3.3): Let A be a real symmetric matrix with zeros on the diagonal. Let d be such that the l₁ norm of each row in A is at most d. Suppose that for every x,y ∈{0,1}ⁿ with ‹x,y›=0. Then the spectral radius of A is O(α(log(d/α)+1)). An interesting consequence of this lemma is a converse to the Expander Mixing Lemma. * This research is supported by the Israeli Ministry of Science and the Israel Science Foundation.     

On disjointly representable sets

A system of setsE ₁,E ₂, ...,E _k ⊂X is said to be disjointly representable if there existx ₁,x ₂, ...,x _k teX such thatx _i teE _j ⇔i=j. Letf(r, k) denote the maximal size of anr-uniform set-system containing nok disjointly representable members. In the first section the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf(r, k),k>3 are established. The last two sections contain some generalizations, in particular we prove an analogue of Sauer’ theorem [16] for uniform set-systems. Dedicated to Paul Erdős on his seventieth birthday     

Canonical edge-colourings of locally finite graphs

A variety of results on edge-colourings are proved, the main one being the following: ifG is a graph without loops or multiple edges and with maximum degree Δ=Δ(G), and if ν is a given integer 1≦ν≦Δ(G), thenG can be given a proper edge-colouring with the coloursc ₁, ...,c _Δ+1 with the additional property that any edge colouredc _μ with μ≧ν is on a vertex which has on it edges coloured with at least ν − 1 ofc ₁, ...,c _v .     

Every large set of equidistant (0, +1, −1)-vectors forms a sunflower

A theorem of Deza asserts that ifH ₁, ...,H _m ares-sets any pair of which intersects in exactlyd elements and ifm ≧s ² −s+2, then theH _i form aΔ-system, i.e. . In other words, every large equidistant (0, 1)-code of constant weight is trivial. We give a (0, +1, −1) analogue of this theorem.     

Sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p ₁³ + ... + p _j ³ with |p _i − (N/j)^1/3| ≦ $ N^{1/3 - \delta _j + \varepsilon } $ N^{1/3 - \delta _j + \varepsilon } (1 ≦ i ≦ j), for δ _j = 1/45; 1/30; 1/25; 2/45, respectively.     

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ 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.     

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.     

The roots of <Emphasis Type="Italic">σ</Emphasis>-polynomials

Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G, where . If σ(G,x) has at least an unreal root, then G is said to be a σ-unreal graph. Let δ(n) be the minimum edgedensity over all n vertices graphs with σ-unreal roots. In this paper, by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et al. is given and the following results are obtained: For any positive integer a and rational number 0≤c≤1, there exists at least a graph sequence {G _i}_1≤i≤a such that G _i is σ-unreal and δ(G _i)→c as n→∞ for all 1 ≤i≤a, and moreover, δ(n)→0 as n→∞. Supported by the National Natural Science Foundation of China (10061003) and the Science Foundation of the State Education Ministry of China.     

On the chromatic number of random geometric graphs

Given independent random points X ₁,...,X _n ∈ℝ ^d with common probability distribution ν, and a positive distance r=r(n)>0, we construct a random geometric graph G _n with vertex set {1,..., n} where distinct i and j are adjacent when ‖X _i −X _j ‖≤r. Here ‖·‖ may be any norm on ℝ ^d , and ν may be any probability distribution on ℝ ^d with a bounded density function. We consider the chromatic number χ(G _n ) of G _n and its relation to the clique number ω(G _n ) as n→∞. Both McDiarmid [11] and Penrose [15] considered the range of r when $r \ll \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$r \ll \left( {\tfrac{{\ln n}} {n}} \right)^{1/d} and the range when $r \gg \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$r \gg \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}, and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the ‘phase change’ range when $r \sim \left( {\tfrac{{t\ln n}} {n}} \right)^{1/d}$r \sim \left( {\tfrac{{t\ln n}} {n}} \right)^{1/d} with t>0 a fixed constant. Both [11] and [15] asked for the behaviour of the chromatic number in this range. We determine constants c(t) such that $\tfrac{{\chi (G_n )}} {{nr^d }} \to c(t)$\tfrac{{\chi (G_n )}} {{nr^d }} \to c(t) almost surely. Further, we find a “sharp threshold” (except for less interesting choices of the norm when the unit ball tiles d-space): there is a constant t ₀>0 such that if t≤t ₀ then $\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$\tfrac{{\chi (G_n )}} {{\omega (G_n )}} tends to 1 almost surely, but if t>t ₀ then $\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$\tfrac{{\chi (G_n )}} {{\omega (G_n )}} tends to a limit >1 almost surely.     

Balancing vectors in the max norm

Let v₁, ..., v _n be vectors inR ⁿ of max norm at most one. It is proven that there exists a choice of signs for which all partial sums have max norm at mostKn ^1/2. It is further shown that such a choice of signs must be anticipatory—there is no way to choose thei-th sign without knowledge of v _j forj>i.     

A characterization of limiting distributions of estimators in an autoregressive process

Summary Let the random variablesX ₁,X ₂, ...,X _n be generated by the first-order autoregressive modelX _i =θX _i−1 +e _i wheree _i ,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ², and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators of θ as well as a local asymptotic minimax bound of the risks of estimators.