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

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

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

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 $\hat{R}_r(H)$, 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 $P_n$, it is known that $\Omega(r^2n)=\hat{R}_r(P_n)=O((r^2\log r)n)$ 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 $P_{n}^{(k)}$; i.e. $\hat{R}_r(P_{n}^{(k)})=O_{r,k}(n)$. Winter gave the first non-trivial lower bounds on the 2-color size-Ramsey number of $P_{n}^{(k)}$ for $k\geq 3$; i.e. $\hat{R}_2(P_{n}^{(3)})\geq \frac{8}{3}n-O(1)$ and $\hat{R}_2(P_{n}^{(k)})\geq \lceil\log_2(k+1)\rceil n-O_k(1)$ for $k\geq 4$. We consider the problem of giving a lower bound on the $r$-color size-Ramsey number of $P_{n}^{(k)}$ (for fixed $k$ and growing $r$). Our main result is that $\hat{R}_r(P_n^{(k)})=\Omega_k(r^kn)$ 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. $\hat{R}_r(P_{k+m}^{(k)})=\Theta_k(r^m)$ for all $1\leq m\leq k$. All of our results generalize to $\ell$-overlapping $k$-uniform paths $P_{n}^{(k, \ell)}$. In particular we note that when $1\leq \ell\leq \frac{k}{2}$, we have $\Omega_k(r^{2}n)=\hat{R}_r(P_{n}^{(k, \ell)})=O((r^2\log r)n)$ which essentially matches the best known bounds for graphs mentioned above. Additionally, in the case $k=3$, $\ell=2$, and $r=2$, we give a more precise estimate which implies $\hat{R}_2(P^{(3)}_{n})\geq \frac{28}{9}n-O(1)$, improving on the above-mentioned lower bound of Winter in the case $k=3$.Comment: 18 pages, updated based on referee comment

On the multicolor Ramsey numbers of balanced double stars

The balanced double star on $2n+2$ vertices, denoted $S_{n,n}$, is the tree obtained by joining the centers of two disjoint stars each having $n$ leaves. Let $R_r(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_N$ there is a monochromatic copy of $G$, and let $R_r^{\mathrm{bip}}(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_{N,N}$ there is a monochromatic copy of $G$. It is known that $R_2(S_{n,n})=3n+2$ and $R_2^{\mathrm{bip}}(S_{n,n})=2n+1$ \cite{HJ}, but very little is known about $R_r(S_{n,n})$ and $R^{\mathrm{bip}}_r(S_{n,n})$ when $r\geq 3$ (other than the bounds which follow from considerations on the number of edges in the majority color class). In this paper we prove the following for all $n\geq 1$ (where the lower bounds are adapted from existing examples): $$(r-1)2n+1\leq R_r(S_{n,n})\leq (r-\frac{1}{2})(2n+2)-1,$$and $$(2r-4)n+1\leq R^{\mathrm{bip}}_r(S_{n,n})\leq (2r-3+\frac{2}{r}+O(\frac{1}{r^2}))n.$$ These bounds are similar to the best known bounds on $R_r(P_{2n+2})$ and $R_r^{\mathrm{bip}}(P_{2n+2})$, where $P_{2n+2}$ is a path on $2n+2$ vertices (which is also a balanced tree). We also give an example which improves the lower bound on $R^{\mathrm{bip}}_r(S_{n,n})$ when $r=3$ and $r=5$

Consensus-based care recommendations for adults with myotonic dystrophy type 1

Purpose of review Myotonic dystrophy type 1 (DM1) is a severe, progressive genetic disease that affects between 1 in 3,000 and 8,000 individuals globally. No evidence-based guideline exists to inform the care of these patients, and most do not have access to multidisciplinary care centers staffed by experienced professionals, creating a clinical care deficit. Recent findings The Myotonic Dystrophy Foundation (MDF) recruited 66 international clinicians experienced in DM1 patient care to develop consensus-based care recommendations. MDF created a 2-step methodology for the project using elements of the Single Text Procedure and the Nominal Group Technique. The process generated a 4-page Quick Reference Guide and a comprehensive, 55-page document that provides clinical care recommendations for 19 discrete body systems and/or care considerations. Summary The resulting recommendations are intended to help standardize and elevate care for this patient population and reduce variability in clinical trial and study environments. Described as “one of the more variable diseases found in medicine,” myotonic dystrophy type 1 (DM1) is an autosomal dominant, triplet-repeat expansion disorder that affects somewhere between 1:3,000 and 1:8,000 individuals worldwide.1 There is a modest association between increased repeat expansion and disease severity, as evidenced by the average age of onset and overall morbidity of the condition. An expansion of over 35 repeats typically indicates an unstable and expanding mutation. An expansion of 50 repeats or higher is consistent with a diagnosis of DM1. DM1 is a multisystem and heterogeneous disease characterized by distal weakness, atrophy, and myotonia, as well as symptoms in the heart, brain, gastrointestinal tract, endocrine, and respiratory systems. Symptoms may occur at any age. The severity of the condition varies widely among affected individuals, even among members of the same family. Comprehensive evidence-based guidelines do not currently exist to guide the treatment of DM1 patients. As a result, the international patient community reports varied levels of care and care quality, and difficulty accessing care adequate to manage their symptoms, unless they have access to multidisciplinary neuromuscular clinics. Consensus-based care recommendations can help standardize and improve the quality of care received by DM1 patients and assist clinicians who may not be familiar with the significant variability, range of symptoms, and severity of the disease. Care recommendations can also improve the landscape for clinical trial success by eliminating some of the inconsistencies in patient care to allow more accurate understanding of the benefit of potential therapies

Measurements of the pp → ZZ production cross section and the Z → 4ℓ branching fraction, and constraints on anomalous triple gauge couplings at √s = 13 TeV

Four-lepton production in proton-proton collisions, pp -> (Z/gamma*)(Z/gamma*) -> 4l, where l = e or mu, is studied at a center-of-mass energy of 13 TeV with the CMS detector at the LHC. The data sample corresponds to an integrated luminosity of 35.9 fb(-1). The ZZ production cross section, sigma(pp -> ZZ) = 17.2 +/- 0.5 (stat) +/- 0.7 (syst) +/- 0.4 (theo) +/- 0.4 (lumi) pb, measured using events with two opposite-sign, same-flavor lepton pairs produced in the mass region 60 4l) = 4.83(-0.22)(+0.23) (stat)(-0.29)(+0.32) (syst) +/- 0.08 (theo) +/- 0.12(lumi) x 10(-6) for events with a four-lepton invariant mass in the range 80 4GeV for all opposite-sign, same-flavor lepton pairs. The results agree with standard model predictions. The invariant mass distribution of the four-lepton system is used to set limits on anomalous ZZZ and ZZ. couplings at 95% confidence level: -0.0012 < f(4)(Z) < 0.0010, -0.0010 < f(5)(Z) < 0.0013, -0.0012 < f(4)(gamma) < 0.0013, -0.0012 < f(5)(gamma) < 0.0013