Constrained Ramsey Numbers

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision

Implications of a scalar dark force for terrestrial experiments

A long-range intergalactic force between dark matter (DM) particles, mediated by an ultralight scalar, is tightly constrained by galactic dynamics and large scale structure formation. We examine the implications of such a ‘‘dark force” for several terrestrial experiments, including Eötvös tests of the Weak Equivalence Principle (WEP), direct-detection DM searches, and collider studies. The presence of a dark force implies a nonvanishing effect in Eötvös tests that could be probed by current and future experiments depending on the DM model. For scalar DM that is a singlet under the standard model gauge groups, a dark force of astrophysically relevant magnitude is ruled out in large regions of parameter space by the DM relic density and WEP constraints. WEP tests also imply constraints on the Higgs-exchange contributions to the spin-independent (SI) DM-nucleus direct-detection cross section. For WIMP scenarios, these considerations constrain Higgs-exchange contributions to the SI cross section to be subleading compared to gauge-boson mediated contributions. In multicomponent DM scenarios, a dark force would preclude large shifts in the rate for Higgs decay to two photons associated with DM-multiplet loops that might otherwise lead to measurable deviations at the LHC or a future linear collider. The combination of observations from galactic dynamics, large scale structure formation, Eötvös experiments, DM-direct-detection experiments, and colliders can further constrain the size of new long-range forces in the dark sector

On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page

Electron-to-Tau Lepton Flavor Violation at the Electron-Ion Collider

We analyze the potential sensitivity of a search for $e\rightarrow\tau$ conversion at a proposed electron-ion collider (EIC) facility. To that end, we calculate the cross sections for $e\rightarrow\tau$ events in a leptoquark framework assuming that the leptoquark masses are on the order of several hundred GeV or more. Given present limits on leptoquarks from direct searches at HERA and rare decay processes, an EIC sensitive to 0.1 fb $e\rightarrow\tau$ cross sections could probe previously unexplored regions of parameter space for these lepton flavor violating events (assuming 90 GeV center-of-mass energy and 10 fb$^{-1}$ integrated luminosity). Depending on the species of leptoquark and flavor structure of the couplings, an EIC search could surpass the HERA and rare process sensitivity to $e\rightarrow\tau$ conversion amplitudes by as much as an order of magnitude or more. We also derive updated limits on quark flavor-diagonal LFV leptoquark interactions using the most recent BaBar $\tau\rightarrow e\gamma$ search. We find that limits from an EIC $e\rightarrow\tau$ search could be competitive with the most recent $\tau\rightarrow e\gamma$ limit for a subset of the quark flavor-diagonal leptoquark couplings. Using an SU(5) GUT model in which leptoquark couplings are constrained by the neutrino masses and mixing, we illustrate how observable leptoquark-induced $e\rightarrow\tau$ conversion can be consistent with stringent LFV limits imposed by $\mu\rightarrow e\gamma$ and $\mu\rightarrow e$ conversion searches.Comment: 19 pages, 7 figure

Exploiting c\mathbf{c}-Closure in Kernelization Algorithms for Graph Problems

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show

A Tight Connection Between Direct and Indirect Detection of Dark Matter through Higgs Portal Couplings to a Hidden Sector

We present a hidden Abelian extension of the Standard Model including a complex scalar as a dark matter candidate and a light scalar acting as a long range force carrier between dark matter particles. The Sommerfeld enhanced annihilation cross-section of the dark matter explains the observed cosmic ray excesses. The light scalar field also gives rise to potentially large cross-sections of dark matter on nucleon, therefore providing an interesting way to probe this model simultaneously at direct and indirect dark matter search experiments. We constrain the parameter-space of the model by taking into account CDMS-II exclusion limit as well as PAMELA and FermiLAT data.Comment: 15 pages, 7 figures. Added discussions on tuning and inverse Compton scattering constraints. References added and updated. Matches the published versio

Higgs CP Violation from Vectorlike Quarks

We explore CP violating aspects in the Higgs sector of models where new vectorlike quarks carry Yukawa couplings mainly to the third generation quarks of the Standard Model. We point out that in the simplest model, Higgs CP violating interactions only exist in the hWW channel. At low energy, we find that rare B decays can place similarly strong constraints as those from electric dipole moments on the source of CP violation. These observations offer a new handle to discriminate from other Higgs CP violating scenarios such as scalar sector extensions of the Standard Model, and imply an interesting future interplay among limits from different experiments.Comment: 19 pages, 5 figure