A. For several models of random constraint satisfaction problems, it was conjectured by physicists and later proved that a sharp satis ability transition occurs. For random k-and related models it happens at clause density α α sat 2 k . Just below the threshold, further results suggest that the solution space has a " " structure of a large bounded number of near-orthogonal clusters inside {0, 1} N .In the unsatis able regime α > α sat , it is natural to consider the problem of max-satis ability: violating the least number of constraints.is is a combinatorial optimization problem on the random energy landscape de ned by the problem instance. For a simpli ed variant, the strong refutation problem, there is strong evidence that an algorithmic transition occurs around α N k/2−1 . For α bounded in N, a very precise estimate of the max-sat value was obtained by Achlioptas, Naor, and Peres (2007), but it is not sharp enough to indicate the nature of the energy landscape. Later work (Sen, 2016; Panchenko, 2016) shows that for α very large (roughly, Ω(64 k )) the max-sat value approaches the mean-eld (complete graph) limit: this is conjectured to have an " " structure where near-optimal con gurations form clusters within clusters, in an ultrametric hierarchy of in nite depth inside {0, 1} N . A stronger form of was shown in several recent works to have algorithmic implications (again, in complete graphs). Consequently we nd it of interest to understand how the model transitions from near α sat , to (conjecturally) for large α. In this paper we show that in the random regular kmodel, the description breaks down already above α 4 k /k 3 . is is proved by an explicit perturbation in the parameter space. e choice of perturbation is inspired by the "bug proliferation" mechanism proposed by physicists (Montanari and Ricci-Tersenghi, 2003;Krzakala, Pagnani, and Weigt, 2004), corresponding roughly to a percolation-like threshold for a subgraph of dependent variables.

A graph G is weakly H-saturated if the complete graph is obtained by iteratively completing copies of H minus an edge. We identify the threshold p c at which the Erdős-Rényi graph G n,p is likely to be weakly H-saturated, for all H such that H \ e is 2-balanced for every edge e ∈ H. The threshold is sharp if this holds strictly. We also establish a general asymptotic lower bound for p c , which holds for all graphs H, and is sharp in many cases. Our results apply for instance when H = K r , solving a problem of Balogh, Bollobás and Morris.

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Tennis as a sport has undergone significant improvement in the past few decades in the former socialist countries. The study is going to introduce the historical background, where exactly East-European Women Tennis Players in large numbers, belonging to the world elite originated from. The article reviews the motivational theories valid in sports and focusing on female players defines the doctrine. The study based on an empirical research which was targeted to reveal the factors behind the motivation of women tennis players. The research was implemented among (n=31) elite women tennis player. Data collection was implemented both by questionnaires and half structured interviews. The outcome of the research will be introduced through 5 dimensions. Satisfaction of East-European elite women tennis players with their (a) career, (b) the amount won at prize money tournaments (c) the relationship between the invested time, money and energy during their sport career and their results, (d) the opinion on the possible restart of their career, (e) the after retirement plans of tennis women players. One of the major consequences of the study is that women tennis players of the world elite have different approach concerning the successfulness of their career, a large percentage of them agree, their prize money won in tournaments they can create material goods for themselves will establishes for their further lives. Tennis today not only provides them with a sporting opportunity, it is also a career model. Their passion for the sport makes many of them work around tennis, even after the end of their players lifeway.

A nemzetközi sportot hosszú évtizedeken keresztül hagyományosan a nyugati nemzetek uralták. A tenisz népszerűsége emelkedett, a társadalmi elit arisztokratikus szórakozásából a középosztály sportjává vált. Ezzel párhuzamosan a sportágban megjelenő lehetőségek is megnőttek, mára milliárdos üzlet lett. A tenisz világszerte az egyik legsikeresebb női sport, a női teniszezők a legjobban fizetett sportolók közé tartoznak. A volt szocialista országok játékosai egyre nagyobb szerepet vállalnak a profi sportággá fejlődött globális teniszben. A tanulmány leírja a jelenség okait, középpontba állítva a már visszavonult és a ma is aktív elit magyar teniszező nőket. A kutatást a már visszavonult (n1=27), illetve az aktív (n2=20) magyar elit teniszezőnők körében tettük meg. Az adatgyűjtést kérdőíves módszerrel végeztük. A statisztikai elemzés igazolja, hogy a női teniszezők sportteljesítményének és sportkultúrájának vizsgálatában a játékosok elégedettségét differenciált tényezők befolyásolják. A tanulmány egyik fő következtetése, hogy a teniszkarrier már életpályamodellként is funkcionál.