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Matt Bowen

Publications and source records attributed to Matt Bowen.

10 recordsLinked to original sources
arXiv2026

Measurable matchings in unbalanced graphs

Let $G$ be a locally finite multigraph that is bipartite and "unbalanced," meaning that it has a nontrivial bipartition $(A,B)$ with $\mathrm{deg}(x) > \mathrm{deg}(y)$ for all $x \in A$ and $y \in B$. We explore matchings in such graphs through the lens of descriptive set theory. In particular, we show that when $G$ is Borel and $\mu$ is a Borel probability measure on its vertex set, there is a Borel matching in $G$ that covers $\mu$-almost every vertex in $A$. This was previously known only under the assumption that $\mu$ is $G$-invariant, which we eliminate using a novel probabilistic approach. We also describe various extra conditions that imply the existence of a Borel matching covering every vertex in $A$. Along the way, we confirm a conjecture of the first and third named authors concerning the existence of Borel independent complete sections in Borel graphs of finite asymptotic separation index. In addition to their intrinsic interest, our results have applications to various other topics, such as edge-colorings, balanced orientations, and equidecomposition theory for group actions. For example, we show that the measurable edge-chromatic number of every Borel multigraph with finite maximum degree $\Delta$ is at most $\lfloor\frac{3\Delta}{2}\rfloor$, matching Shannon's optimal bound for finite multigraphs. Another example is that paradoxical Borel group actions with finite asymptotic separation index admit paradoxical decompositions with Borel pieces. This refines a result of Marks and Unger.

math.LO↗
arXiv2025

Measurable one-ended spanning trees

We show that a one-ended, locally finite, measurable graph on a standard probability space admits a measurable one-ended spanning subtree if and only if it is measure-hyperfinite. This answers a question posed by Bowen, Poulin, and Zomback and extends recent results of Tim\'ar and Conley, Gaboriau, Marks, and Tucker-Drob.

math.LO↗
arXiv2024

Measurable Regular Subgraphs

We show that every $d$-regular bipartite Borel graph admits a Baire measurable $k$-regular spanning subgraph if and only if $d$ is odd or $k$ is even. This gives the first example of a locally checkable coloring problem which is known to have a Baire measurable solution on Borel graphs but not a computable solution on highly computable graphs. We also prove the analogous result in the measure setting for hyperfinite graphs.

math.LO↗
arXiv2024

Monochromatic non-commuting products

We show that a finite coloring of an amenable group contains `many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables. This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem. Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.

math.CO↗
arXiv2022

Composite Ramsey theorems via trees

We prove a theorem ensuring that the compositions of certain Ramsey families are still Ramsey. As an application, we show that in any finite coloring of $\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set $B$ with $(A+B)\cup (AB)$ monochromatic, addressing a problem of Kra, Moreira, Richter, and Robertson. In fact, we prove an iterated version of this result, which ensures the existence of monochromatic patterns such as $\{a\circ_1 (b \circ_2 c): \circ_i\in \{+,\cdot\}\}, $ generalizing a Ramsey theorem of Bergelson and Moreira that was previously only known to hold for colorings of $\mathbb{Q}$ rather than colorings of $\mathbb{N}$. Our main new technique is an extension of the color focusing method that involves trees rather than sequences.

math.CO↗
arXiv2022

Monochromatic products and sums in $2$-colorings of $\mathbb{N}$

We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\},$ and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i: i\leq k\}$ for any $k\in\mathbb{N}.$ Along the way we prove a monochromatic products of sums theorem that extends Hindman's theorem and a colorful variant of this result that holds in any 'balanced' coloring.

math.CO↗
arXiv2021

Definable Kőnig theorems

Let $X$ be a Polish space with Borel probability measure $μ,$ and let $G$ be a Borel graph on $X$ with no odd cycles and maximum degree $Δ(G).$ We show that the Baire measurable edge chromatic number of $G$ is at most $Δ(G)+1$, and if $G$ is $μ$-hyperfinite then the $μ$-measurable edge chromatic number obeys the same bound. More generally, we show that $G$ has Borel edge chromatic number at most $Δ(G)$ plus its asymptotic separation index.

math.LO↗
arXiv2021

Colored unavoidable patterns and balanceable graphs

We study a Turán-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $Ω(n^{2-1/tr^r})$ edges in each color contains a member from certain finite family $\mathcal{F}_t^r$ of $r$-edge-colored complete graphs. We conjecture that $Ω(n^{2-1/t})$ edges in each color are sufficient to find a member from ${\mathcal{F}}_t^r$. A result of Girão and Narayanan confirms this conjecture when $r=2$. Next, we study a related problem where the corresponding Turán threshold is linear. We call an edge-coloring of a path $P_{rk}$ balanced if each color appears $k$ times in the coloring. We show that any $3$-edge-coloring of a large complete graph with $kn+o(n)$ edges in each color contains a balanced $P_{3k}$. This is tight up to a constant factor of $2$. For more colors, the problem becomes surprisingly more delicate. Already for $r=7$, we show that even $n^{2-o(1)}$ edges from each color does not guarantee existence of a balanced $P_{7k}$.

math.CO↗