On a geometric statement of Ramsey type

A simple geometric assertion of Ramsey type, concerning families of straight lines in the Euclidean space R 3 \mathbb{R}^{3} , is formulated, and it is shown that the assertion turns out to be undecidable within the framework of ZFC set theory.

Phase Variation Measurement in Mach–Zehnder Interferometric Switch

Phase variations of the interrogation field lead to frequency shifts in Ramsey-type atomic clocks. This paper reports the development of a 300 MHz Mach–Zehnder (MZ) switch that effectively suppresses phase-transient effects. Similar to MZ interferometers, this radio-frequency (RF) MZ switch comprises two arms that are power- and phase-matched with each other. By inserting a PIN diode RF switch in one arm, the other arm remains undisturbed, freeing it of the phase transient. Trigger phase fluctuation measurements are implemented by using a lock-in amplifier to extract the in-phase and quadrature (I/Q) demodulation data. The results show that the extinction ratio of the RF MZ switch phase fluctuations is <5 μrad, which is significantly lower than that of a PIN (50 μrad). When applied to a cesium fountain clock, the RF MZ switch produces a frequency shift better than 1.73 × 10−16.

Complementarity Modeling of a Ramsey-Type Equilibrium Problem with Heterogeneous Agents

We contribute to the field of Ramsey-type equilibrium models with heterogeneous agents. To this end, we state such a model in a time-continuous and time-discrete form, which in the latter case leads to a finite-dimensional mixed complementarity problem. We prove the existence of solutions of the latter problem using the theory of variational inequalities and present further properties of its solutions. Finally, we compute the growth dynamics in a calibrated model in which households differ with respect to their relative risk aversion, their discount factors, their initial wealth, and with respect to their interest rates on savings.

Ramsey-type numbers involving graphs and hypergraphs with large girth

Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle C k . The existence of such graphs H was confirmed by the third author and Ruciński. We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(C k ; r) is the r-colour Ramsey number for the cycle C k ) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic C k Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

Open Problems Column Edited by William Gasarch This Issue's Column!

In this column we state a class of open problems in Ramsey Theory. The general theme is to take Ramsey-type statements that are false and weaken them by allowing the homogenous set to use more than one color. This concept is not new, and the theorems we state and/or prove are not new; however, the open questions that request easier proofs of the known theorems (or weaker versions) may be new. We use the phrase an elementary proof. This is not meant to be a technical or rigorous term. What we really mean is a proof that can be taught in an undergraduate combinatorics course. A good example of what we mean is the proof of Theorem 9.3.

The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets and Chains

Motivated by the paper, Boolean lattices: Ramsey properties and embeddings Order, 34 (2) (2017), of Axenovich and Walzer, we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring of elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$ as an induced subposet. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also derive some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.

Finding Unavoidable Colorful Patterns in Multicolored Graphs

We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollobás conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and Montágh. Here, we obtain a characterization for arbitrary values of $r$, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre.

On Colorful Edge Triples in Edge-Colored Complete Graphs

An edge-coloring of the complete graph $$K_n$$ K n we call F-caring if it leaves no F-subgraph of $$K_n$$ K n monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when $$F=K_{1,3}$$ F = K 1 , 3 and $$F=P_4$$ F = P 4 we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring of $$K_n$$ K n . An explicit family of $$2\lceil \log _2 n\rceil $$ 2 ⌈ log 2 n ⌉ 3-edge-colorings of $$K_n$$ K n so that every quadruple of its vertices contains a totally multicolored $$P_4$$ P 4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the cycle of length 5.