Given two graphs G and H , an H -matching of G (or a tiling of G with H) is a subgraph of G consisting of vertex-disjoint copies of H . For an r-chromatic graph H on h vertices, we write u = u(H) for the smallest possible color-class size in any r-coloring of H . The critical chromatic number of H is the number cr(H)= (r− 1)h=(h− u). A conjecture of Koml# os states that for every graph H , ther...

This is a survey of the author's recent results on the Kadison and Halmos similarity problems and the closely connected notion of " length " of an operator algebra. We start by recalling a well known conjecture formulated by Kadison [Ka] in 1955. Kadison's similarity problem. Let A be a unital C *-algebra and let u: A → B(H) (H Hilbert) be a unital homomorphism (i.e. we have u(1) = 1 and u(ab) ...

The Erdős-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n, where δ(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty b...

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g G$. First, we prove that if is $n$ >31\psi(C_n)/77$, $C_n$ cyclic $n$, then supersolvable. This proves conjecture M.~{T\u{a}rn\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi Maj put forward following conjecture: If $H\leq G$, \leqslant \psi(H) |G:H|^2$. In sequel, by an example show this not sat...

We study pointwise convergence of the solutions to Schrödinger equations with initial datum f ∈ H(R). The conjecture is that the solution ef converges to f almost everywhere for all f ∈ H(R) if and only if s ≥ 1/4. The conjecture is known true for one spatial dimension and the convergence when s > 1/2 was verified for n ≥ 2. Recently, concrete progresses have been made in R for some s < 1/2. Ho...

A version of the Erdős-Hajnal conjecture for tournaments states that for every tournament H every tournament T that does not contain H as a subtournament, contains a transitive subtournament of size at least n (H) for some (H) > 0, where n is the order of T . For any fixed tournament H we can denote by n0(H) the supremum over all ≥ 0 satisfying the following statement: every tournament T of ord...

Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group whose socle is a Mathieu group such that cd(G) = cd(H), then there exists an abelian subgroup A of G such that G/A is isomorphic to H. In view of Huppert’s conjecture (2000), we also provide some examples to show that...

A graph is H-free if it has no induced subgraph isomorphic to H. An old conjecture of Conlon, Sudakov and the second author asserts that: • For every graph H, there exists ε > 0 such that in every H-free graph with n > 1 vertices there are two disjoint sets of vertices, of sizes at least εn and εn, complete or anticomplete to each other. This is equivalent to: • The ‘sparse linear conjecture’: ...