arXiv subjects

Shengjie Xie

Publications and source records attributed to Shengjie Xie.

6 recordsLinked to original sources

arXiv2026

Hypergraphs without Subgraphs of Given Connectivity

In this paper, we study the problem of determining the maximum number $F_r(n,k)$ of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case, initiated by Mader, is a classical problem in graph theory that remains open. We first establish a limit theorem for $F_r(n,k)$ for all $k\ge r\ge 2$. As a consequence, we prove for the first time that, in Mader's problem (i.e., $r=2$), there exists a constant $c_k>0$ such that $F_2(n,k)=c_k n+O_k(1)$, and, for every $r\ge 3$, we determine $F_r(n,k)$ up to an $O(n)$ error term, thereby identifying its leading asymptotic term. We also address a related question of Carmesin by establishing a tight bound for $r$-uniform hypergraphs with no $(k+1)$-connected subgraph on more than $Ck$ vertices for any constant $C>2$ and sufficiently large $r$, and further obtain an asymptotically tight bound in the case $C=2$. Our proof combines the separator tree method introduced by Carmesin with several new combinatorial and optimization techniques, and we conclude with related remarks and open problems.

math.CO↗

arXiv2026

An exponential improvement for Ramsey lower bounds

We prove a new lower bound on the Ramsey number $r(\ell, C\ell)$ for any constant $C > 1$ and sufficiently large $\ell$, showing that there exists $\varepsilon=\varepsilon(C)> 0$ such that \[ r(\ell, C\ell) \geq \left(p_C^{-1/2} + \varepsilon\right)^\ell, \] where $p_C \in (0, 1/2)$ is the unique solution to $C = \frac{\log p_C}{\log(1 - p_C)}$. This provides the first exponential improvement over the classical lower bound obtained by Erdős in 1947.

math.CO↗

arXiv2024

Discrepancies of perfect matchings in hypergraphs

In this paper, we determine the minimum degree threshold of perfect matchings with high discrepancy in $r$-edge-colored $k$-uniform hypergraphs for all $k\geq 3$ and $r\geq 2$, thereby completing the investigation into discrepancies of perfect matchings that has recently attracted significant attention. Our approach identifies this discrepancy threshold with a novel family of multicolored uniform hypergraphs and reveals new phenomena not covered in previous studies. In particular, our results address a question of Balogh, Treglown and Zárate-Guerén concerning 3-uniform hypergraphs.

math.CO↗

arXiv2024

Finding irregular subgraphs via local adjustments

For a graph $H$, let $m(H,k)$ denote the number of vertices of degree $k$ in $H$. A conjecture of Alon and Wei states that for any $d\geq 3$, every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying $|m(H,k)-\frac{n}{d+1}|\leq 2$ for every $0\leq k \leq d$. This holds easily when $d\leq 2$. An asymptotic version of this conjecture was initially established by Frieze, Gould, Karoński and Pfender, subsequently improved by Alon and Wei, and most recently enhanced by Fox, Luo and Pham, approaching its complete range. All of these approaches relied on probabilistic methods. In this paper, we provide a novel framework to study this conjecture, based on localized deterministic techniques which we call local adjustments. We prove two main results. Firstly, we show that every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying $|m(H,k)-\frac{n}{d+1}|\leq 2d^2$ for all $0\leq k \leq d$, which provides the first bound independent of the value of $n$. Secondly, we confirm the case $d=3$ of the Alon-Wei Conjecture in a strong form. Both results can be generalized to multigraphs and yield efficient algorithms for finding the desired subgraphs $H$. Furthermore, we explore a generalization of the Alon-Wei Conjecture for multigraphs and its connection to the Faudree-Lehel Conjecture concerning irregularity strength.

math.CO↗

arXiv2022

On-Chip High Extinction Ratio Single-Stage Mach-Zehnder Interferometer based on Multimode Interferometer

On-chip high extinction ratio Mach-Zehnder interferometers (MZI) have always attracted interest from researchers as it can be used in many applications in astrophotonics, optical switching, programmable photonic circuits, and quantum information. However, in previous research studies, ultra-high extinction ratio on-chip MZIs have only been achieved by using a multi-stage MZI approach. In this paper, we investigate a high extinction ratio single-stage MZI based on two cascaded multimode interferometers (MMI). We determine that TM noise is an important factor that can prevent us from achieving a high extinction ratio MZI. By introducing a bend-based TM filter without additional loss, we experimentally demonstrate that such a TM filter can improve the maximum extinction ratio of the MMI-MZI by more than 10 dB. With the TM filter, we report a record high 61.2 dB extinction ratio in a single stage, thermally tunable MMI-MZI with only 1.5 dB insertion loss and more than 60nm bandwidth. These results pave the way for many interesting applications.

physics.optics↗

arXiv2020

Integrated Arbitrary Filter with Spiral Gratings: Design and Characterization

We report the design and characterization of a high performance integrated arbitrary filter from 1450 nm to 1640 nm. The filter's target spectrum is chosen to suppress the night-sky OH emission lines, which is critical for ground-based astronomical telescopes. This type of filter is featured by its large spectral range, high rejection ratio and narrow notch width. Traditionally it is only successfully accomplished with fiber Bragg gratings. The technique we demonstrate here is proven to be very efficient for on-chip platforms, which can bring many benefits for device footprint, performance and cost. For the design part, two inverse scattering algorithms are compared, the frequency domain discrete layer-peeling (f-DLP) and the time domain discrete layer-peeling (t-DLP). f-DLP is found to be superior for the grating reconstruction in terms of accuracy and robustness. A method is proposed to resolve the non-uniformity issue caused by the non-zero layer size in the DLP algorithm. The designed 55-notch filter is 50-mm-long and implemented on a compact Si3N4/SiO2 spiral waveguide with a total length of 63 mm. Experimentally, we demonstrate that the device has a insertion loss as low as 2.5 dB, and that the waveguide propagation loss is as low as 0.10 dB/cm. We are also able to achieve uniform notch depths and 3-dB widths of about 28 dB and 0.22 nm, respectively.

astro-ph.IM↗