Laudal-type theorems in PN

Laudal Type Theorems for Algebraic Curves

Laudal’s Lemma states that if C is an integral curve in P3 of degree d > s2 + 1 and Z is its general plane section, then C is contained in a surface of degree s provided that Z is contained in a curve of degree s. The aim of this paper is to extend Laudal’s Lemma to possibly reducible curves proving that, under the unavoidable hypothesis that the Hilbert function of the generic plane section is...

Common Fixed Point Theorems of Integral Type in Modular Spaces

Pn, Pn and Pn in Lines of Action

The endgame of Lines of Action (LOA) is problematic because (1) LOA evaluation functions are not good predictors in the case of forced wins and (2) the state-space complexity of the LOA endgame is too large to use endgame databases. Because of the frequency of forced moves and the mobility component in the LOA endgame the utility of proof-number (PN) search is investigated in this paper. Also t...

Tolerance in Helly-Type Theorems

In this paper we introduce the notion of tolerance in connection with Helly type theorems and prove, using the Erdős-Gallai theorem, that any Helly type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number of exceptional sets or points. In particular, we analyze some of the classical Helly type theorems, such as Caratheodory’s and Tverberg’s theorems,...

On Ramsey-type theorems

In [21], Frank Plumpton Ramsey proved what has turned out to be a remarkable and important theorem which is now known as Ramsey's theorem. This result is a generalization of the pigeonhole principle and can now be seen as part of a family of theorems of the same flavour. These Ramsey-type theorems all have the common feature that they assert, in some precise combinatorial way, that if we deal w...