We study subtyping of recursive types in the presence of associative and commutative products—that is, subtyping modulo a restricted form of type isomorphisms. We show that this relation, which we claim is useful in practice, is a composition of the usual subtyping relation with the recently proposed notion of equality up to associativity and commutativity of products, and we propose an efficie...

We give explicit versions of Helfgott’s Growth Theorem for SL2, as well as of the Bourgain-Gamburd argument for expansion of Cayley graphs modulo primes of subgroups of SL2(Z) which are Zariski-dense in SL2.

A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

Let G = (V,E) be a graph of order n. A distance magic labeling of G is a bijection l : V → {1, 2, . . . , n} for which there exists a positive integer k such that ∑ x∈N(v) l(x) = k for all v ∈ V , where N(v) is the open neighborhood of v. In this paper we deal with circulant graphs C(1, p). The circulant graph Cn(1, p) is the graph on the vertex set V = {x0, x1, . . . , xn−1} with edges (xi, xi...

We consider embeddings of structures which preserve spectra: if g : M → S with S computable, then M should have the same spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph G. Such structures are...

Given a prime p, we say that S ⊆ Z/pZ is a critical set of density d if and only if |S| = dp and S has the least number of arithmetic progressions among all other sets S having at least dp elements. In this context, an arithemtic progression is a triple of residue classes n, n+m,n+2m modulo p. Note that this includes “trivial” progressions, which are ones wherem ≡ 0 (mod p), as well as “non-tri...

The circular graph C(n, m) is such a graph that whose vertex set is {v0, v1, v2, · · · , vn−1} and edge set is {vivi+1, vivi+m | i = 0, 1, · · · , n − 1}, where m,n are natural numbers, addition is modulo n, and 2 ≤ m ≤ b2 c. This paper shows the crossing number of the circular graph C(2m + 2,m)(m ≥ 3) is m + 1.

In this paper we examine the orders of vertex-transitive self-complementary uniform hypergraphs. In particular, we prove that if there exists a vertex-transitive selfcomplementary k-uniform hypergraph of order n, where k = 2 or k = 2 + 1 and n ≡ 1 (mod 2), then the highest power of any prime dividing n must be congruent to 1 modulo 2. We show that this necessary condition is also sufficient in ...

A proper edge-coloring of a graph G with colors 1, , t  is called an interval cyclic t  coloring if all colors are used, and the edges incident to each vertex ( ) v V G  are colored with ( ) G d v consecutive colors by modulo t , where ( ) G d v is the degree of the vertex v in G . In this paper some properties of interval cyclic edge-colorings are investigated. Also, we give some bounds for...

Alon, Kleitman, Lipton, Meshulam, Rabin and Spencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph F = fF1; F2; : : : ; Fd(q 1)+1g, where q is a primepower, and d denotes the maximal degree of the hypergraph, there exists an F0 F , such that jSF2F0 F j 0 (mod q). We give a direct, alternative proof for this theorem, and we also give an explicit construction of a hyperg...