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In this paper, we shed new light on the spectrum of the relation algebra we call $A_{n}$, which is obtained by splitting the non-flexible diversity atom of $6_{7}$ into $n$ symmetric atoms. Precisely, we show that the minimum value in $\text{Spec}(A_{n})$ is at most $2n^{6 + o(1)}$, which is the first polynomial bound and improves upon the previous bound due to Dodd \& Hirsch (\textit{J. Relational Methods in Computer Science} 2013). We also improve the lower bound to $2n^{2} + 4n + 1$, which is asymptotically double the trivial bound of $n^{2} + 2n + 3$. In the process, we obtain stronger results regarding $\text{Spec}(A_{2}) =\text{Spec}(32_{65})$. Namely, we show that $1024$ is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results., 17 pages

In this paper, we consider relational structures arising from Comer's finite field construction, where the cosets need not be sum free. These Comer schemes generalize the notion of a Ramsey scheme and may be of independent interest. As an application, we give the first finite representation of $34_{65}$. We complement our upper bounds with some lower bounds. Using a SAT solver, we establish that neither $33_{65}$ nor $34_{65}$ are representable on fewer than $24$ points.

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras $33_{37}, 35_{37}$, $77_{83}$, $78_{83}$, $80_{83}$, $82_{83}$, $83_{83}$, $1310_{1316}$, $1313_{1316}$, $1315_{1316}$, and $1316_{1316}$. Prior to our paper, only $83_{83}$ and $1316_{1316}$ were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed (anti-)Ramsey scheme. The notion of a directed (anti-)Ramsey scheme may be of independent interest. We complement our upper bounds with some lower bounds. Namely, we show that any square representation of $31_{37}$ requires at least $16$ points, any square representation of $33_{37}$ requires at least $11$ points, and any square representation of $35_{37}$ requires at least $14$ points. Our technique adapts previous work of Alm, et. al. (Algebra Universalis 2022), in that we examine the combinatorial structure induced by the flexible atom.

We develop a random model for relation algebras. We prove some preliminary results and pose questions that lay out a new direction of research.

In Alm-Hirsch-Maddux (2016), relation algebras $\mathfrak{L}(q,n)$ were defined that generalize Roger Lyndon's relation algebras from projective lines, so that $\mathfrak{L}(q,0)$ is a Lyndon algebra. In that paper, it was shown that if $q>2304n^2+1$, $\mathfrak{L}(q,n)$ is representable, and if $q, 8 pages, 2 figures

The chromatic number $��\left(\mathcal{E^2}\right)$ of the plane is known to be some integer between 4 and 7, inclusive. We prove a limiting result that says, roughly, that one cannot increase the lower bound on $��\left(\mathcal{E^2}\right)$ by pasting Moser Spindles together, even countably many., To appear in Australasian Journal of Combinatorics

In this paper we study Ramsey numbers for trees of diameter 3 (bistars) vs., respectively, trees of diameter 2 (stars), complete graphs, and many complete graphs. In the case of bistars vs. many complete graphs, we determine this number exactly as a function of the Ramsey number for the complete graphs. We also determine the order of growth of the bipartite $k$-color Ramsey number for a bistar., Comment: 8 pages, 6 figures

We generalize a result of Tibor Gallai as follows: for any finite set of points $\mathcal{S}$ in the plane, if the plane is colored in finitely many colors, then there exist $2^{\aleph_0}$ monochromatic subsets of the plane homothetic to $\mathcal{S}$. Furthermore, we prove an even stronger result for $n$-dimensional Euclidean space., Comment: 7 pages; To appear in Journal of Combinatorics

We consider a modification of Winkler's "dots and coins" problem, where we constrain the dots to lie on a square lattice in the plane. We construct packings of "coins" (closed unit disks) using motif patterns., Comment: 10 pages, 11 figures

In this note we consider the question of how large ` can be so that M` fits inside Sub(G) as a 0-1 sublattice, for a finite abelian group G. We answer the question for G = Zn × Zn: in this case ` = p + 1, where p is the smallest prime dividing n.

For positive integers $m$ and $n$, define $f(m,n)$ to be the smallest integer such that any subset $A$ of the $m \times n$ integer grid with $|A| \geq f(m,n)$ contains a rectangle; that is, there are $x\in [m]$ and $y \in [n]$ and $d_{1},d_{2} \in \mathbb{Z}^{+}$ such that all four points $(x,y)$, $(x+d_{1},y)$, $(x,y+d_{2})$, and $(x+d_{1},y+d_{2})$ are contained in $A$. In \cite{kovarisosturan}, K��vari, S��s, and Tur��n showed that $\dlim_{k \to \infty}\dfrac{f(k,k)}{k^{3/2}} = 1$. They also showed that whenever $p$ is a prime number, $f(p^{2},p^{2}+p) = p^{2}(p+1)+1$. We recover their asymptotic result and strengthen the second, providing cleaner proofs which exploit a connection to projective planes, first noticed by Mendelsohn in \cite{mendelsohn87}. We also provide an explicit lower bound for $f(k,k)$ which holds for all $k$., 9 pages