626. [K1. 2. a.] Note on Note 605 (Gazette, xi. p. 20)

A Note on P-sets

A P-set is a set S of positive integers such that no element of S divides the sum of any two (not necessarily being different) larger elements. Erdös and Sárközy [2] conjectured that there exists a constant c > 0 such that for every P-set S we have |{s ∈ S : s ≤ N}| < N1−c for infinitely many integers N . For P-sets S consisting of pairwise coprime integers this conjecture has been proved by T....

A note on p-curvatures

In this note, we give an arithmetic criterion for the Lie-irreducibility of linear differential equations based on p-curvatures.

2 1 Se p 20 05 A Note on the Solvablity of Groups ∗

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K * M . We call the group M ∩ K/L a c-section of M . And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M . For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L2(p) or Zq, where p and q are primes, ...

2 7 Se p 20 06 A Note on ( 3 , 1 ) ∗ - Choosable Toroidal Graphs †

An (L, d)-coloring is a mapping φ that assigns a color φ(v) ∈ L(v) to each vertex v ∈ V (G) such that at most d neighbors of v receive colore φ(v). A graph is called (m, d)-choosable, if G admits an (L, d)-coloring for every list assignment L with |L(v)| ≥ m for all v ∈ V (G). In this note, it is proved that every toroidal graph, which contains no adjacent triangles and contains no 6-cycles and...

A note on potentially K1, 1, t-graphic sequences