Landing in the future: Biological experiments on Earth and in space orbit

The development of an Earth biosatellite to duplicate the parameters of pressure, temperature, humidity and others in a space environment onboard Cosmos-1129 is discussed. Effects of a space environment on fruit flies, dogs, laboratory rats in procreation, behavior, stress, biorhythm, body composition, gravitation preference, and cell cultures are examined. The space environment for agricultural products is also studied. The effects of heavy nuclei of galactic space radiation on biological objects inside and outside the satellite is studied, and methods of electrostatic protection are developed

Calculating Ramsey Numbers by Partitioning Colored Graphs

In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey Numbers exactly. The partitioning theorem we prove is that for k ≥ 1, in every edge colouring of Kn with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring of Kn is connected in red, we prove a stronger result—that it is possible to cover all the vertices with k red paths and a blue balanced complete (k + 2)-partite graph. Using these results we determine the Ramsey Number of a path, Pn, versus a balanced complete t-partite graph on tm vertices, Kt m, whenever m ≡ 1 (mod n−1). We show that in this case R(Pn, Kt m) = (t − 1)(n − 1) + t(m − 1) + 1, generalizing a result of Erd˝os who proved the m = 1 case of this result. We also determine the Ramsey Number of a path Pn versus the power of a path P t n . We show that R(Pn, Pt n ) = t(n − 1) + j n t+1 k , solving a conjecture of Allen, Brightwell, and Skokan

Rainbow matchings and rainbow connectedness

Aharoni and Berger conjectured that every collection of n matchings of size n+1 in a bipartite graph contains a rainbow matching of size n. This conjecture is related to several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. The conjecture is known to hold when the matchings are much larger than n + 1. The best bound is currently due to Aharoni, Kotlar, and Ziv who proved the conjecture when the matchings are of size at least 3n/2 + 1. When the matchings are all edge-disjoint and perfect, the best result follows from a theorem of H¨aggkvist and Johansson which implies the conjecture when the matchings have size at least n + o(n). In this paper we show that the conjecture is true when the matchings have size n + o(n) and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least φn + o(n) where φ ≈ 1.618 is the Golden Ratio. Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest

Growth of Graph Powers

For a graph G, its rth power is constructed by placing an edge between two vertices if they are within distance r of each other. In this note we study the amount of edges added to a graph by taking its rth power. In particular we obtain that, for r ≥ 3, either the rth power is complete or "many" new edges are added. In this direction, Hegarty showed that there is a constant ε > 0 such e(G3) ≥ (1 + ε)e(G). We extend this result in two directions. We give an alternative proof of Hegarty's result with an improved constant of ε = 1/6. We also show that for general

An approximate version of a conjecture of Aharoni and Berger

Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by n colours with at least n+1 edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding problem of Brualdi and Stein about transversals in Latin squares. Here an approximate version of the AharoniBerger Conjecture is proved—it is shown that if there are at least n + o(n) edges of each colour in a proper n-edge-colouring of a bipartite multigraph then there is a rainbow matching using every colour

Linearly many rainbow trees in properly edge-coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjectur

A COUNTEREXAMPLE TO STEIN'S EQUI-n-SQUARE CONJECTURE

In 1975 Stein conjectured that in every n × n array filled with the numbers 1, . . . , n with every number occuring exactly n times, there is a partial transversal of size n−1. In this note we show that this conjecture is false by constructing such arrays without partial transverals of size n − 1/ 42 ln n

Edge-disjoint rainbow trees in properly coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. We discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. The main result which we discuss is that in every proper edge-colouring of Kn there are 10−6n edge-disjoint isomorphic spanning rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method it is also possible to show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint spanning rainbow trees, giving a further improvement on the Brualdi-Hollingsworth Conjectur

Ramsey goodness of cycles

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) \geq (| G| - 1)(\chi (H) - 1) + \sigma (H), where \chi (H) is the chromatic number of H and \sigma (H) is the size of the smallest color class in a \chi (H)-coloring of H. A graph G is called H-good if R(G, H) = (| G| - 1)(\chi (H) - 1) + \sigma (H). The notion of Ramsey goodness was introduced by Burr and Erd\H os in 1983 and has been extensively studied since then. In this paper we show that if n \geq 1060| H| and \sigma (H) \geq \chi (H) 22, then the n-vertex cycle Cn is H-good. For graphs H with high \chi (H) and \sigma (H), this proves in a strong form a conjecture of Allen, Brightwell, and Skokan

Ramsey goodness of paths

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)- coloring of H. A graph G is called H-good if R(G, H) = (|G| − 1)(χ(H) − 1) + σ(H). The notion of Ramsey goodness was introduced by Burr and Erd˝os in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n ≥ 4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan