Yury Person

WEAK QUASI-RANDOMNESS FOR UNIFORM HYPERGRAPHS

We study quasi-random properties of k-uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will nd several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung-Graham-Wilson theorem for quasi-random graphs. Moreover, let Kk be the complete graph on k vertices and M(k) the line graph of the graph of the k-dimensional hypercube. We will show that the pair of graphs (Kk;M(k)) has the property that if the number of copies of both Kk and M(k) in another graph G are as expected in the random graph of density d, then G is quasi-random (in the sense of the Chung-Graham-Wilson theorem) with density close to d.

On extremal hypergraphs for Hamiltonian cycles

We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Turan- and Dirac-type results. While the Turan-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields a sufficient condition relying solely on the minimum vertex degree.

On colourings of hypergraphs without monochromatic fano planes

For k-uniform hypergraphs F and H and an integer r, let cr,F(H) denote the number of r-colourings of the set of hyperedges of H with no monochromatic copy of F, and let

Exact Results on the Number of Restricted Edge Colorings for Some Families of Linear Hypergraphs

c_{r,F}(n)=max_{HinccHn} c_{r,F}(H)

Minimum H-decompositions of graphs: Edge-critical case

, where the maximum runs over all k-uniform hypergraphs on n vertices. Moreover, let ex(n,F) be the usual extremal or Turan function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F. n nFor complete graphs F = Kl and r = 2, Erdős and Rothschild conjectured that c2,Kl(n) = 2ex(n,Kl). This conjecture was proved by Yuster for l = 3 and by Alon, Balogh, Keevash and Sudakov for arbitrary l. In this paper, we consider the question for hypergraphs and show that, in the special case when F is the Fano plane and r = 2 or 3, then cr,F(n) = rex(n,F), while cr,F(n) ≫ rex(n,F) for r ≥ 4.

A regularity lemma and twins in words

For k-uniform hypergraphs F and H and an integer , let denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let , where the maximum is taken over the family of all k-uniform hypergraphs on n vertices. Moreover, let be the usual extremal function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F. Here, we consider the question for determining for F being the k-uniform expanded, complete graph or the k-uniform Fan(k)-hypergraph with core of size , where , and we show n n n n n n nfor and n large enough. Moreover, for or , for k-uniform hypergraphs H on n vertices, the equality only holds if H is isomorphic to the l-partite, k-uniform Turan hypergraph on n vertices, once n is large enough. On the other hand, we show that is exponentially larger than , if .

Note on strong refutation algorithms for random k-SAT formulas

For a given graph H let @fH(n) be the maximum number of parts that are needed to partition the edge set of any graph on n vertices such that every member of the partition is either a single edge or it is isomorphic to H. Pikhurko and Sousa conjectured that @fH(n)=ex(n,H) for @g(H)>=3 and all sufficiently large n, where ex(n,H) denotes the maximum size of a graph on n vertices not containing H as a subgraph. In this article, their conjecture is verified for all edge-critical graphs. Furthermore, it is shown that the graphs maximizing @fH(n) are (@g(H)-1)-partite Turan graphs.

A randomized version of Ramsey's theorem

For a word S, let f(S) be the largest integer m such that there are two disjoint identical (scattered) subwords of length m. Let f(n,@S)=min{f(S):S is of length n, over alphabet @S}. Here, it is shown that2f(n,{0,1})=n-o(n) using the regularity lemma for words. In other words, any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length o(n). A similar result is proven for k identical subwords of a word over an alphabet with at most k letters.

An expected polynomial time algorithm for coloring 2-colorable 3-graphs

Abstract We present a simple strong refutation algorithm for random k-SAT formulas. Our algorithm applies to random k-SAT formulas on n variables with ω ( n ) n ( k + 1 ) / 2 clauses for any ω ( n ) → ∞ . In contrast to the earlier results of Coja-Oghlan, Goerdt, and Lanka (for k = 3 , 4) and Coja-Oghlan, Cooper, and Frieze (for k ⩾ 5 ), which address the same problem for even sparser formulas our algorithm is more elementary.

Highly connected coloured subgraphs via the regularity lemma

The standard randomization of Ramseys theorem asks for a fixed graph F and a fixed number r of colors: for what densities p = p(n) can we asymptotically almost surely color the edges of the random graph G(n, p) with r colors without creating a monochromatic copy of F. This question was solved in full generality by Rodl and Rucinski [Combinatorics, Paul Erdős is eighty, vol. 1, 1993, 317–346; J Am Math Soc 8(1995), 917–942]. In this paper we consider a different randomization that was recently suggested by Allen et al. [Random Struct Algorithms, in press]. Let documentclass{article} usepackage{amsmath,amsfonts} pagestyle{empty} begin{document}