A cycle in an edge-colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge-colored graph G, define S(G) = {n ≥ 2 | no n-cycle of G is rainbow}. Then S(G) is a monoid with respect to the operation n ◦m = n + m − 2, and thus there is a least positive integer π(G), the period of S(G), such that S(G) contains the arithmetic progression {N + kπ(...

We present two proofs, one proof-theoretic and one model-theoretic, showing that adding the ß ^-collection axioms to any bounded first-order theory R of arithmetic yields an extension which is V ̂ -conservative over R. Preliminaries. A theory of arithmetic R contains the nonlogic symbols 0, S, +, ■, and < . R may contain further nonlogical symbols; in particular S2 is a theory of arithmetic [1]....

A symmetric encryption scheme allows Alice and Bob to privately exchange a sequence of messages in the presence of an eavesdropper Eve. We will assume that Alice and Bob share a random secret key K. How Alice and Bob managed to share a key without the adversary’s knowledge is not going to be our concern here. The encryption scheme consists of an encryption algorithm E that takes as input the ke...

Let N be a positive integer. In 1964, Roth [3], see also [4] and [5], proved that no matter how we partition {1, . . . , N} into two sets there will always be an arithmetic progression which contains a preponderance of terms from one of the two sets. In particular, let ε1, . . . , εN be elements of {1,−1} and put εi = 0 for i < 1 and i > N . He proved that there exist positive numbers c0 and c1...