More on quasi-random graphs, subgraph counts and graph limits

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases. The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.Comment: 35 page

Útburkolatok összgamma-sugárzásának vizsgálata

The world average of radioactive background radiation is 2.4 mSv pro year. The main part of this background radiation is come from builded environment. From this reason many international project was focused on the radioctive radiation of the building materials. However the radiation from pavemant materials has been less researched. Therefore our departement feel it important to adress more reseach of radiation of different type of pavement and layers of pavement. The ND-497 type portable scintillation detector was used for our measurements. During our measurements the radiation of pavement surface and the layers of pavement was measured directly. The results of our project was that the radioactive radiation over pavement was driven by the materials porosity and the type of this materials similar to the building materials. It is beacuse in case of low porosity the radon - comes from earth and pavement - diffusion was blocked. It means that the pavements are decrease the radiation compare to the base soil results from low radioactivity of concrete and high porosity of pavement. In case of asphalt the situation is a bit complicated. The dose is decreasing over the sidewalk but increase over the roadway compare the base soil. In this case we have to remember that the asphalt are build over the concrete base. Finally the ratio of this two layer will define the final radiaoctive radiation

On restricted colourings of Kn

The authors investigate Ramsey-type extremal problems for finite graphs. In Section 1, anti-Ramsey numbers for paths are determined. For positive integers k and n let r=f(n,Pk) be the maximal integer such that there exists an edge colouring of Kn using precisely r colours but not containing any coloured path on k vertices with all edges having different colors. It is shown that f(n,P2k+3+ε)=t⋅n−(t+12)+1+ε for t≥5, n>c⋅t2 and ε=0,1. In Section 2, K3-spectra of colourings are determined. Given S⊆{1,2,3}, the authors investigate for which r and n there exist edge colourings of Kn using precisely r colours such that all triangles are s-coloured for some s∈S and, conversely, every s∈S occurs. Section 3 contains suggestions for further research

A hierarchy of randomness for graphs

AbstractIn this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness.The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems