A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0 0$, as $n$ tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.

Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels

In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erd\H{o}s on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum $d$-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl

Novel PAMAM-PEG-Peptide Conjugates for siRNA Delivery Targeted to the Transferrin and Epidermal Growth Factor Receptors

The transferrin (TfR) and epidermal growth factor receptors (EGFR) are known to be overexpressed on the surface of a wide variety of tumor cells. Therefore, the peptides B6 (TfR specific) and GE11 (targeted to the EGFR) were linked to the PAMAM (polyamidoamine) structure via a polyethylenglycol (PEG) 2 kDa chain with the aim of improving the silencing capacity of the PAMAM-based dendriplexes. The complexes showed an excellent binding capacity to the siRNA with a maximal condensation at nitrogen/phosphate (N/P) 2. The nanoparticles formed exhibited hydrodynamic diameters below 200 nm. The zeta potential was always positive, despite the complexes containing the PEG chain in the structure showing a drop of the values due to the shielding effect. The gene silencing capacity was assayed in HeLa and LS174T cells stably transfected with the eGFPLuc cassette. The dendriplexes containing a specific anti luciferase siRNA, assayed at different N/P ratios, were able to mediate a mean decrease of the luciferase expression values of 14% for HeLa and 20% in LS174T cells, compared to an unspecific siRNA-control. (p < 0.05). In all the conditions assayed, dendriplexes resulted to be non-toxic and viability was always above 75%