A subset S of a poset P is called a k-family if S is the union of k or fewer antichains of P. Greene and Kleitman showed that the set of k-families of a poset forms a locally distributive lattice. Kung, Rota, and Yan asked whether every finite locally distributive lattice can be represented in this way. It is shown here that the answer is no.
A set B of positive integers is defined as a Sidon set if all differences b−b′ are distinct, where b,b′∈B and b≠b′. By confirming a problem posed by Lev at CANT 2004, Cilleruelo and Nathanson (2008) [2] obtained a nice result as follows: for every Sidon set B and every function ω(x)→∞, there exists a set A of positive integers such that every positive integer can be uniquely expressed as a−a′, where a,a′∈A and A(x)≥B(x/3)−ω(x) for all sufficiently large x. In this paper, we improve the coefficient from 1/3 to 1/2, that is: for every Sidon set B and every positive function ω(x)→∞, there exists a set A of positive integers such that every positive integer can be uniquely expressed as a−a′, where a,a′∈A andB(x/2)−ω(x)≤A(x)≤B(x/2)+ω(x) for all x≥1.
Generalized Cayley maps are embeddings of Cayley graphs into closed surfaces, either orientable or non-orientable, admitting a group of automorphisms that acts regularly on the vertices. When the surface is orientable and the automorphism group preserves orientation, these embeddings, known as Cayley maps, have been extensively studied. A seminal result by Conder and Tucker classifies orientably-regular Cayley maps on cyclic groups. This paper completes the classification of regular generalized Cayley maps on cyclic groups by addressing the remaining cases: non-orientable surfaces or orientable surfaces with the regular cyclic group acting in an orientation-reversing manner.
Let A be a set of nonnegative integers. We call A an asymptotic basis of order 2 if every sufficiently large integer can be expressed as the sum of two elements in A; otherwise, A is referred to as an asymptotic nonbasis of order 2. An asymptotic basis A of order 2 is minimal if the set A∖{a} is no longer an asymptotic basis of order 2 for every a∈A. Correspondingly, an asymptotic nonbasis A of order 2 is maximal if the set A∪{b} becomes an asymptotic basis of order 2 for every nonnegative integer b∉A. For a set A of integers, we define A(n)=|A∩[1,n]|. Let k be a positive integer. In this paper, we prove that there exists a maximal asymptotic nonbasis A={ai}i=1∞ satisfying n/2≤A(n)≤n/2+(k−1)/2 for any nonnegative integer n and limsupi→∞(ai+1−ai)=k with k≥2; we also prove that there exists a minimal asymptotic basis B={bi}i=1∞ satisfying n/2≤B(n)≤n/2+(k−1)/2 for any nonnegative integer n and limsupi→∞(bi+1−bi)=k with k≥3. These results answer two problems put forward by Tang and Chen.
