Upper density of monochromatic infinite paths

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

Kulish, Mykola Hurovych [КУЛІШ МИКОЛА ГУРОВИЧ] (1892–1937)

Mykola Hurovych Kulish was born on December 5, 1892 (Old Style; December 18 New Style) in Chaplinka, Tavricheskaia gubernia in the Russian Empire (today Ukraine’s Kherson oblast) to a peasant family. After his mother’s death, he left home to attend school in nearby Oleshky, where he met his future wife, Antonina, and his lifelong friend, Ivan Shevchenko (literary pseudonym, Dniprovskii). He started university in Odessa in 1914, but was soon conscripted into the Russian Imperial Army, and fought on the Smolensk front. After the February Revolution he served in the frontline soldiers’ committees, and he continued to fight with the Red Army during the Civil War. He joined the Communist Party in 1919. In 1922 Kulish was decommissioned to the post of school inspector in the People’s Commissariat of Enlightenment (Narkomos) in Odessa, where he began to write seriously. He joined the Odessa branch of the organization Hart [Tempering], headed the Zinovievsk (Russian imperial Ekaterinoslav, today’s Kirovohrad) branch of the Party’s literary journal Chervonyishliakh [Red Path], and in 1925 the Soviet Ukrainian party-state promoted Kulish to Kharkiv, then the capital of Soviet Ukraine.

British Philosophy in the Mid-Century

In the summer of 1953 a lecture-course organized by the British Council was given at Peterhouse, Cambridge. The Faculty of Moral Science were responsible for the programme of lectures and discussions, and Miss Margaret Master man and Dr. Theodore Red path were appointed by the Faculty as joint directors. The lectures must have been well received by the teachers of philosophy who attended and participated in the discussions— representatives from the Continent, the United States and even China were on hand; and the suggestion arose quite naturally that they be published in a single volume. However, some of the lecturers wished to redraft the papers they had read and hence the essays now presented, under the editorship of C. A. Mace, in the volume British Philosophy in the Mid-Century, A Cambridge Symposium (George Allen and Unwin Ltd., 1957) are in a number of instances dressed up and greatly expanded versions of the lectures actually given. Further, it is worth noting that the essay by G. E. Moore was written specifically for this volume and is based on a discussion held with some students who attended the sessions at Peterhouse. Thus it is that the papers now published vary greatly in length from the welcome ten pages contributed by G. E. Moore to the seventy-nine pages from Miss Master man. This disparity is, of course, an anomaly in a volume explicitly designed to reflect trends in very recent British philosophy and to convey, to those relatively unfamiliar with it, some reliable picture of its condition at mid-century. But this volume records a present-day Cambridge symposium and as such it must in some measure reflect quite local interests and conditions. However, most of the contributors are very well known and the essays now presented to the public contain a good deal that will interest and profit the reader.