Sums, products, and exponents in two-colorings of the naturals
We prove that for any coloring of the naturals using two colors there are monochromatic sets of the form $\{x,y,xy,x+iy:i\leq k\}$ and $\{x,y,x^y,xy^i:i\leq k\}$ for any $k$.
math.CO↗
arXiv subjects
Matthew Bowen
Publications and source records attributed to Matthew Bowen.
The uniform Gardner conjecture and rounding Borel flows
We study groups which satisfy Gardner's equidecomposition conjecture for uniformly distributed sets. We prove that an amenable group has this property if and only if it does not admit $(\mathbb{Z}/2\mathbb{Z}) *(\mathbb{Z}/2\mathbb{Z})$ as a quotient by a finite subgroup. Our technical contribution is an algorithm for rounding Borel flows for actions of amenable groups.
math.LO↗
One-ended spanning trees and definable combinatorics
Let $(X,τ)$ be a Polish space with Borel probability measure $μ,$ and $G$ a locally finite one-ended Borel graph on $X.$ We show that $G$ admits a Borel one-ended spanning tree generically. If $G$ is induced by a free Borel action of an amenable (resp., polynomial growth) group then we show the same result $μ$-a.e. (resp., everywhere). Our results generalize recent work of Timár, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved this in the probability measure preserving setting. We apply our theorem to find Borel orientations in even degree graphs and measurable and Baire measurable perfect matchings in regular bipartite graphs, refining theorems that were previously only known to hold for measure preserving graphs. In particular, we prove that bipartite one-ended $d$-regular Borel graphs admit Baire measurable perfect matchings.
math.CO↗
Perfect matchings in hyperfinite graphings
We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite bipartite graphing admits a measurable perfect matching if it is one-ended or the degree is odd. We give several applications of this result, answering various open questions in the field. For instance, we extend the Lyons--Nazarov theorem by characterizing bipartite Cayley graphs which admit a factor of iid perfect matching, answering the bipartite case of a well-known question of Lyons and Nazarov, popularized by Kechris and Marks. Moreover, we show how our results apply to measurable equidecompositions and, in particular, generalize the recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring. Our approach applies more generally to rounding measurable perfect fractional matchings.
math.CO↗
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 $χ$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $χ$ $\varepsilon$-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 $χ$ 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.
math.CO↗
The Sprague-Grundy function for some selective compound games
We analyze the Sprague-Grundy functions for a class of almost disjoint selective compound games played on Nim heaps. Surprisingly, we find that these functions behave chaotically for smaller Sprague-Grundy values of each component game yet predictably when any one heap is sufficiently large.
math.CO↗
Narrow Trans-TeV Higgs Bosons and $H\to hh$ Decays: Two LHC Search Paths for a Hidden Sector Higgs Boson
We consider the addition of a condensing singlet scalar field to the Standard Model. Such a scenario may be motivated by any number of theoretical ideas, including the common result in string-inspired model building of singlet scalar fields charged under some hidden sector gauge symmetry. For concreteness, we specify an example model of this type, and consider the relevant constraints on Higgs physics, such as triviality, perturbative unitarity and precision electroweak analysis. We then show that there are two unique features of the phenomenology that present opportunities for discovery at the Large Hadron Collider (LHC). First, it is possible to identify and discover a narrow trans-TeV Higgs boson in this scenario -- a mass scale that is well above the scale at which it is meaningful to discuss a SM Higgs boson. Second, the decays of the heavier scalar state into the lighter Higgs bosons can proceed at a high rate and may be the first discovery mode in the Higgs sector.
hep-ph↗