Note on a Partition Function Which Assumes All Integral Values

Note on the Derivative and the Variation of a Function Depending on All the Values of Another Function

Odd values of the partition function

Let p(n) denote the number of partitions of an integer n. Recently the author has shown that in any arithmetic progression r (mod t), there exist infinitely many N for which p(N) is even, and there are infinitely many M for which p(M) is odd, provided there is at least one such M. Here we construct finite sets of integers Mi for which p(Mi) is odd for an odd number of i. Whereas Euler’s recurre...

On a Total Function Which Overtakes All Total Recursive Functions. *

We discuss here a function that is part of the folklore around the P =?N P problem. This function g * is defined over all time–polynomial Turing machines; we trivially change it into a g which is defined over all Turing machines, and show that g overtakes all total recursive functions. We then show that g is dominated by a generalized Busy Beaver function.

A Note on Tree-partition-width

A tree-partition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartition-width of G is the minimum number of vertices in a bag in a tree-partition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with tree-width k ≥ 3 and maximum degree ∆ ≥ 1...

A Note on Strong Negative Partition Relations

We analyze a natural function definable from a scale at a singular cardinal, and provide a fairly simple proof that there is a strong connection between failures of square brackets partition relations at sucessors of singular cardinals and the club–guessing ideal idp(C̄, Ī).