introduction: knee osteoarthritis (ko) is the most common joint disease for which there is no optimal treatment. monochromatic infrared photo energy (mipe) is a relatively new light modality used to reduce pain and increase circulation. low level laser therapy (lllt) is another light modality used to reduce pain in ko. methods: the aim of this study was to compare the effects of the mipe and ll...

This paper is motivated by a recent result of Wolf [12] on the minimum number of monochromatic 4-term arithmetic progressions (4-APs, for short) in Zp, where p is a prime number. Wolf proved that there is a 2-coloring of Zp with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construct...

In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the monochromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that eve...

Let f(2, 3, 4) denote the smallest integer n such that there exists a K4-free graph of order n for which any 2-coloring of its edges yields at least one monochromatic triangle. It is well-known that such a number must exist. For a long time the best known upper bound, provided by J. Spencer, said that f(2, 3, 4) < 3 · 10. Recently, L. Lu announced that f(2, 3, 4) < 10 000. In this note, we will...

Network models allow one to deal with massive data sets using some standard concepts from graph theory. Understanding and investigating the structural properties of a certain data set is a crucial task in many practical applications of network optimization. Recently, labeled network optimization over colored graphs has been extensively studied. Given a (not necessarily properly) edge-colored gr...