The extremal function for partial bipartite tilings

For a fixed bipartite graph H and given number c, 0<c<1, we determine the threshold T_H(c) which guarantees that any n-vertex graph with at edge density at least T_H(c) contains $(1-o(1))c/v(H) n$ vertex-disjoint copies of H. In the proof we use a variant of a technique developed by Komlos~\bcolor{[Combinatorica 20 (2000), 203-218}]Comment: 10 page

Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for publication in Journal of Combinatorial Theory, series

Strong approximation of fractional Brownian motion by moving averages of simple random walks

The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to B.B. Mandelbrot and J.W. van Ness (1968) as a self-similar Gaussian process \WH (t) with stationary increments. Here self-similarity means that (a^{-H}\WH(at): t \ge 0) \stackrel{d}{=} (\WH(t): t \ge 0), where $H\in (0, 1)$ is the Hurst parameter of fractional Brownian motion. F.B. Knight gave a construction of ordinary Brownian motion as a limit of simple random walks in 1961. Later his method was simplified by P. R\'ev\'esz (1990) and then by the present author (1996). This approach is quite natural and elementary, and as such, can be extended to more general situations. Based on this, here we use moving averages of a suitable nested sequence of simple random walks that almost surely uniformly converge to fractional Brownian motion on compacts when H \in (\quart , 1). The rate of convergence proved in this case is O(N^{-\min(H-\quart,\quart)}\log N), where $N$ is the number of steps used for the approximation. If the more accurate (but also more intricate) Koml\'os, Major, Tusn\'ady (1975, 1976) approximation is used instead to embed random walks into ordinary Brownian motion, then the same type of moving averages almost surely uniformly converge to fractional Brownian motion on compacts for any $H \in (0, 1)$. Moreover, the convergence rate is conjectured to be the best possible $O(N^{-H}\log N)$, though only O(N^{-\min(H,\half)}\log N) is proved here.Comment: 30 pages, 4 figure

Cycles are strongly Ramsey-unsaturated

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp who also proved that cycles (except for $C_4$) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle $C_n$, unless n is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph H is obtained by adding a linear number of chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large. This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method