SoCalDM2025

Jonathan Davidson (Cal State LA) - A Combinatorial Design Approach to a Multicolor Bipartite Ramsey Problem 

The multicolor $d$-partite Ramsey problem asks for the sizes of the parts of a complete $d$-uniform $d$-partite graph that guarantees any edge coloring with $c$ colors has a monochromatic $K^{(d)}_{2,\dots,2}$ subgraph. By examining the incidence matrix, the problem may be reformulated in terms of digital geometry. In this setting, the problem generalizes to finding monochromatic vertices of boxes in $d$-dimensional grids. We find an upper bound on the $d$-dimensional problem in terms of the $(d-1)$-dimensional problem. By reformulating the problem again into graph decompositions, we were able to use different types of resolvable designs for constructions that achieve the upper bound in the 2-dimensional case. We also improve some previous bounds for the 3-dimensional case that were based on the Lovasz Local Lemma. 

Lenny Fukshansky (Claremont McKenna College) - On a new absolute version of Siegel’s lemma 

The classical Siegel's lemma (1929) asserts the existence of a nontrivial integer solution to an underdetermined integer homogeneous linear system, whose "size" is small as compared to the size of the coefficients of the system. Far-reaching generalizations of this theorem, producing a full basis for the solution space, were obtained over number fields by Bombieri & Vaaler (1983), and over the field of algebraic numbers by Roy & Thunder (1996), where the "size" was measured by a height function. We obtain a new version of Siegel's lemma, bridging the Bombieri & Vaaler and Roy & Thunder results in two ways: (1) our basis lies over a fixed number field as in Bombieri & Vaaler's theorem; (2) our height-bound does not depend on the number field in question as in Roy & Thunder's theorem. Our result does not imply the previously established ones and is not implied by them, and our basis has some additional interesting properties. Our method is quite different from the previous ones, using only linear algebra. Joint work with Max Forst. 

Mason Shurman (UCI) - Covering random digraphs with Hamilton cycles 

A covering of a digraph $D$ by Hamilton cycles is a collection of directed Hamilton cycles (not necessarily edge-disjoint) that together cover all the edges of $D$. We prove that for $1/2 \geq p\geq \frac{\log^{20} n}{n}$, the random digraph $D_{n,p}$ typically admits an optimal Hamilton cycle covering. Specifically, the edges of $D_{n,p}$ can be covered by a family of $t$ Hamilton cycles, where $t$ is the maximum of the in-degree and out-degree of the vertices in $D_{n,p}$. Notably, $t$ is the best possible bound, and our assumption on $p$ is optimal up to a polylogarithmic factor. 

Claire Levaillant (USC) - Solutions to the Diophantine equation $\sum_{i=1}^n\frac{1}{x_i}=1$ in integers of the form p^a.q^b with p and q two distinct primes  

We present an algorithm to exhibit all the solutions of the Diophantine equation $\sum_{i=1}^n\frac{1}{x_i}=1$ in integers of the form p^a.q^b with p and q two distinct primes, when bounding the exponent of one of the two primes. This problem relates to finding arithmetical structures on the complete graph.

Sehun Jeong (Claremont Graduate University) - Integral quadratic forms and lattice angles 

Let us consider angles between two vectors in the integer lattice Z^n. Fixing one such possible angle and one integer vector x, is there always another integer vector y that makes this angle with x? Assuming that x makes a given angle with some vector, how can we find the shortest such vector y? What if we designate a forbidden set of vectors, what is the shortest y making a given angle with x outside of this forbidden set? It turns out that all of these questions can be reformulated in terms of a search for integer zeros of integral quadratic forms, a rich arithmetic theory. I will discuss this research direction and show results by several authors, then present my recent joint work with Lenny Fukshansky. 

Justin Troyka (Cal State LA) - Growth rates of permutations with given descent or peak set 

Given a set $I \subseteq \mathbb{N}$, consider the sequences $\{d_n(I)\},\{p_n(I)\}$ where for any $n$, $d_n(I)$ and $p_n(I)$ respectively count the number of permutations in the symmetric group $S_n$ whose descent set (respectively peak set) is $I \cap [n-1]$. We investigate the growth rates $\operatorname{gr} d_n(I) = \lim_{n \to \infty} \left(d_n(I)/n!\right)^{1/n}$ and $\operatorname{gr} p_n(I) = \lim_{n \to \infty} \left(p_n(I)/n!\right)^{1/n}$ over all $I \subseteq \mathbb{N}$. Our main contributions are two-fold. Firstly, we prove that the numbers $\operatorname{gr} d_n(I)$ over all $I \subseteq \mathbb{N}$ are exactly the interval $\left[0,2/\pi\right]$. To do so, we construct an algorithm that explicitly builds $I$ for any desired limit $L$ in the interval. Secondly, we prove the numbers $\operatorname{gr} p_n(I)$ for periodic sets $I \subseteq \mathbb{N}$ form a dense set in $\left[0,1/\sqrt[3]{3}\right]$. We do this by explicitly finding, for any prescribed limit $L$ in the interval, a set $I$ whose corresponding growth rate is arbitrarily close to $L$. This talk is based on joint work with Mohamed Omar. 

Yizhe Zhu (USC) - CLTs for linear spectral statistics of inhomogeneous random graphs 

We establish central limit theorems (CLTs) for the linear spectral statistics of the adjacency matrix of inhomogeneous random graphs, under the assumption that the connecting probability profile of the random graphs converges to a graphon limit. Two types of CLTs are derived for the adjacency matrix and the centered adjacency matrix across all sparsity regimes.