We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group Fn 2 . A triangle in Fn 2 is a triple (x, y, z) such that x + y + z = 0. The triangle removal lemma for Fn 2 states that for every ε > 0 there is a δ > 0, such that if a subset A of Fn 2 requires the removal of at least ε · 2n elements to make it triangle-free, then it must contain at l...

We give a new, combinatorial proof for the necklace splitting problem for two thieves using only Tucker's lemma (a combinatorial version of the Borsuk-Ulam theorem). We show how this method can be applied to obtain a related recent result of Simonyi and even generalize it. 1 Necklace Splitting This paper was inspired by the combinatorial proof of Matou2ek [7] of the Lovász-Kneser theorem [6]. H...

A 1993 result of J. Llibre, and M. Misiurewicz, (Theorem A [5]), states that if a continuous map f of a graph into itself has an s-horseshoe, then the topological entropy of f is greater than or equal to logs, that is h( f ) ? logs. Also a 1980 result of L.S. Block, J. Guckenheimer, M. Misiurewicz and L.S. Young (Lemma 1.5 [3]) states that if G is an A-graph of f then h(G) ? h( f ). In this pap...