A digraph is n-unavoidable if it contained in every tournament of order n. We first prove that arborescence n with k leaves (n+k−1)-unavoidable. then oriented tree (n≥2) (32n+32k−2)-unavoidable and (92n−52k−92)-unavoidable, thus (218n−4716)-unavoidable. Finally, we (n+144k2−280k+124)-unavoidable.

Kleene’s formalization FIM of intuitionistic analysis ([3] and [2]) includes bar induction, countable and continuous choice, but is consistent with the statement that there are no non-recursive functions ([5]). Veldman ([12]) showed that in FIM the constructive analytical hierarchy collapses at Σ 2 . These are serious obstructions to interpreting the constructive content of classical analysis, ...

A set of words X is called unavoidable on a given alphabet A if every infinite word on A has a factor in X. For k, q ≥ 1, let c(k, q) be the number of conjugacy classes of words of length k on q letters. An unavoidable set of words of length k on q symbols has at least c(k, q) elements. We show that for any k, q ≥ 1 there exists an unavoidable set of words of length k on q symbols having c(k, q...

Goal-oriented behavior is usually portrayed to be directly related to incentive values. Underlying mechanisms are thought to include reward-oriented response bias and perceptual sensitivity. A recent electrophysiological study by Minamimoto et al. challenges this view, and reports neural activity in the thalamus that appears to counteract response bias. This type of activity might be crucial fo...

A natural generalization of graph Ramsey theory is the study of unavoidable subgraphs in large colored graphs. In this paper, we find a minimal family of unavoidable graphs in two-edgecolored graphs. Namely, for a positive even integer k, let Sk be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint Kk/2s or simply one Kk/2. Bollobás conject...

In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is deened to be the minimum number n such that there is a sequence of graphs G = G 0 , G 1 , : : : , G n , where G i is obtained by contracting one edge in or deleting one edge from each block of G i?1 , and where G n is edgeless. We show that a 3-connected graph has large type if and only if it...

Sharing and maintaining long, random keys is one of the central problems in cryptography. This thesis provides about ensuring the security of a cryptographic key when partial information about it has been, or must be, leaked to an adversary. We consider two basic approaches: 1. Extracting a new, shorter, secret key from one that has been partially compromised. Specifically, we study the use of ...

With the increase in jogging and distance running as a means of attaining and maintaining fitness, and in the light of the difficulty in advising certain groups of patients about their capacity for such exercise, we report on several potentially avoidable factors in the death of a 45 year old Sheffield Marathon runner in June 1983.