Partitioning random graphs into monochromatic components

Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every $r$-colored complete graph can be partitioned into at most $r-1$ monochromatic components; this is a strengthening of a conjecture of Lov\'asz (1975) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into $r$ monochromatic components is possible for sufficiently large $r$-colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if $p\ge \left(\frac{27\log n}{n}\right)^{1/3}$, then a.a.s. in every $2$-coloring of $G(n,p)$ there exists a partition into two monochromatic components, and for $r\geq 2$ if $p\ll \left(\frac{r\log n}{n}\right)^{1/r}$, then a.a.s. there exists an $r$-coloring of $G(n,p)$ such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gy\'arf\'as (1977) about large monochromatic components in $r$-colored complete graphs. We show that if $p=\frac{\omega(1)}{n}$, then a.a.s. in every $r$-coloring of $G(n,p)$ there exists a monochromatic component of order at least $(1-o(1))\frac{n}{r-1}$.Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics Volume 24, Issue 1 (2017) Paper #P1.1

Large monochromatic components in expansive hypergraphs

A result of Gy\'arf\'as exactly determines the size of a largest monochromatic component in an arbitrary $r$-coloring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $r-1\leq k\leq r$. We prove a result which says that if one replaces $K_n^k$ in Gy\'arf\'as' theorem by any ``expansive'' $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform hypergraph $H$ on $n$ vertices in which in which $e(V_1, \dots, V_k)>0$ for all disjoint sets $V_1, \dots, V_k\subseteq V(H)$ with $|V_i|>\alpha$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gy\'arf\'as' result is equivalent to the dual problem of determining the smallest maximum degree of an arbitrary $r$-partite $r$-uniform hypergraph with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \dots, E_k\subseteq E(H)$ with $|E_i|>\alpha$ for all $i\in [k]$ there exists $e_i\in E_i$ for all $i\in [k]$ such that $e_1\cap \dots \cap e_k\neq \emptyset$, then the maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$). We prove our results in this dual setting.Comment: 18 page

A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs

The r-color size-Ramsey number of a k-uniform hypergraph H, denoted by R^r(H), is the minimum number of edges in a k-uniform hypergraph G such that for every r-coloring of the edges of G there exists a monochromatic copy of H. In the case of 2-uniform paths Pn, it is known that Ω(r2n)=R^r(Pn)=O((r2logr)n) with the best bounds essentially due to Krivelevich. In a recent breakthrough result, Letzter, Pokrovskiy, and Yepremyan gave a linear upper bound on the r-color size-Ramsey number of the k-uniform tight path P(k)n; i.e. R^r(P(k)n)=Or,k(n). Winter gave the first non-trivial lower bounds on the 2-color size-Ramsey number of P(k)n for k≥3; i.e. R^2(P(3)n)≥8/3n−O(1) and R^2(P(k)n)≥⌈log2(k+1)⌉n−Ok(1) for k≥4.We consider the problem of giving a lower bound on the r-color size-Ramsey number of P(k)n (for fixed k and growing r). Our main result is that R^r(P(k)n)=Ωk(rkn) which generalizes the best known lower bound for graphs mentioned above. One of the key elements of our proof is a determination of the correct order of magnitude of the r-color size-Ramsey number of every sufficiently short tight path; i.e. R^r(P(k)k+m) = Θk(rm) for all 1≤m≤k.All of our results generalize to ℓ-overlapping k-uniform paths P(k,ℓ)n. In particular we note that when 1≤ℓ≤k/2, we have Ωk(r2n)=R^r(P(k,ℓ)n)=O((r2logr)n) which essentially matches the best known bounds for graphs mentioned above. Additionally, in the case k=3, ℓ=2, and r=2, we give a more precise estimate which implies R^2(P(3)n)≥28/9n−O(1), improving on the above-mentioned lower bound of Winter in the case k=3

Prophylactic balloon angioplasty fails to prolong the patency of expanded polytetrafluoroethylene arteriovenous grafts: Results of a prospective randomized study

AbstractPurpose: Maintenance of hemodialysis access grafts represents an enormous social and clinical problem. Current grafts and graft salvage techniques are inadequate. Consequently, there has been increasing interest in the use of minimally invasive catheter techniques to prophylactically treat stenoses in functioning arteriovenous grafts. Prophylactic balloon angioplasty has been widely suggested as prolonging assisted primary patency. We have performed a prospective randomized trial to compare patients who underwent percutaneous transluminal angioplasty (PTA) for graft stenoses >50% with a control group that received no intervention. Our hypothesis was that to be efficacious a minimal benefit of 20% prolongation in patency would be necessary.Methods: Color flow duplex scanning was used to detect >50% stenoses in functioning expanded polytetrafluoroethylene grafts. Patients were then subjected to confirmatory angiographic evaluation. Those who had angiographic stenoses >50% were randomized to balloon angioplasty or observation. Patients were followed-up with duplex scanning every 2 months. Statistical analysis was performed using the Kaplan-Meier technique. Although demographically the patient groups were well matched, there were more prior interventions and concurrent central stenoses in the treatment group. Outcomes were graft thrombosis, graft dysfunction that precluded dialysis, and six or more PTA procedures within 18 months.Results: In the treatment and observation groups, the 6-month patency rates were 69% ± 7% and 70% ± 7%, respectively. The 12-month patency rates for the treatment and observation groups were 51% ± 6% and 47% ± 4%, respectively. There was no significant difference between these two groups ( p = 0.97), with an 80% confidence limit for detection of a difference greater than 20%.Conclusions: This study demonstrates that a generic approach of PTA to treat all polytetrafluoroethylene grafts with stenoses >50% does not prolong patency and cannot be supported

In-plane magnetic field effect on the neutron spin resonance in optimally doped FeSe0.4_{0.4}Te0.6_{0.6} and BaFe1.9_{1.9}Ni0.1_{0.1}As2_{2} superconductors

We use inelastic neutron scattering to study the effect of an in-plane magnetic field on the magnetic resonance in optimally doped superconductors FeSe$_{0.4}$Te$_{0.6}$ ($T_c=14$ K) and BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$ ($T_c=20$ K). While the magnetic field up to 14.5 Tesla does not change the energy of the resonance, it particially suppresses $T_c$ and the corresponding superconductivity-induced intensity gain of the mode. However, we find no direct evidence for the field-induced spin-1 Zeeman splitting of the resonance. Therefore, it is still unclear if the resonance is the long-sought singlet-triplet excitation directly coupled to the superconducting electron Cooper pairs.Comment: 5 pages, 4 figures, The first two wrong figures are correcte

Flexible and Intelligent Learning Architectures for SOS (FILA-SoS)

Multi-faceted systems of the future will entail complex logic and reasoning with many levels of reasoning in intricate arrangement. The organization of these systems involves a web of connections and demonstrates self-driven adaptability. They are designed for autonomy and may exhibit emergent behavior that can be visualized. Our quest continues to handle complexities, design and operate these systems. The challenge in Complex Adaptive Systems design is to design an organized complexity that will allow a system to achieve its goals. This report attempts to push the boundaries of research in complexity, by identifying challenges and opportunities. Complex adaptive system-of-systems (CASoS) approach is developed to handle this huge uncertainty in socio-technical systems

Electrochemical alkene azidocyanation via 1,4-nitrile migration

An electrochemical method for the azidocyanation of alkenes via 1,4-nitrile migration has been developed. This organic oxidant free method is applicable across various alkene containing cyanohydrins, and provides access to a broad range of synthetically useful 1,2-azidonitriles (28 examples). This methodology was extended to an electrochemical alkene sulfonylcyanation procedure, as well as to access a trifunctionalized hexanenitrile from a malononitrile starting material. The orthogonal derivatization of the products was also demonstrated through chemoselective transformations

Spectroscopy, MOST Photometry, and Interferometry of MWC 314: Is it an LBV or an interacting binary?

MWC 314 is a bright candidate luminous blue variable that resides in a fairly close binary system, with an orbital period of 60.753$\pm$0.003 d. We observed MWC 314 with a combination of optical spectroscopy, broad-band ground- and space-based photometry, as well as with long baseline, near-infrared interferometry. We have revised the single-lined spectroscopic orbit and explored the photometric variability. The orbital light curve displays two minima each orbit that can be partially explained in terms of the tidal distortion of the primary that occurs around the time of periastron. The emission lines in the system are often double-peaked and stationary in their kinematics, indicative of a circumbinary disc. We find that the stellar wind or circumbinary disc is partially resolved in the K\prime-band with the longest baselines of the CHARA Array. From this analysis, we provide a simple, qualitative model in an attempt to explain the observations. From the assumption of Roche Lobe overflow and tidal synchronisation at periastron, we estimate the component masses to be M1 $\approx 5$ M$_\odot$ and M2$\approx 15$ M$_\odot$, which indicates a mass of the LBV that is extremely low. In addition to the orbital modulation, we discovered two pulsational modes with the MOST satellite. These modes are easily supported by a low-mass hydrogen-poor star, but cannot be easily supported by a star with the parameters of an LBV. The combination of these results provides evidence that the primary star was likely never a normal LBV, but rather is the product of binary interactions. As such, this system presents opportunities for studying mass-transfer and binary evolution with many observational techniques.Comment: 26 pages, 7 figures, 5 tables, 2 appendices with 7 additional tables and 2 additional figures. Accepted for publication in MNRA