I started writing this blog post some months ago. Occasion was that my paper “A construction for clique-free pseudorandom graphs” (in joint work Anurag Bishnoi and Valentina Pepe) was accepted by Combinatorica with minor revisions. More precisely, one of the referees was unfavorable of publication because he got the impressions that we are simply restating […]
Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36, 10, 4, 2), but there are 32548 non-isomorphic graphs with parameters (36, 15, 6, 6). Peter Cameron, Random Strongly Regular Graphs? This a shorter version of this report which I just […]
Recently, I collected a short list of phrases for conjectures on a well-known social media platform and several people contributed to it. One can easily find more examples online, but I like my list, so I will keep it here and include references (as far as I have them). Probably, I will add more entries […]
Let $ {X}$ be the orthogonality graph, that is the graph with $ {\{ -1, 1 \}^n}$ as vertices with two vertices adjacent if they are orthogonal. So $ {x, y \in \{ -1, 1 \}^n}$ are adjacent if $ {x \cdot y = x_1y_1 + x_2y_2 + \ldots + x_ny_n = 0}$. There are […]
As Anurag Bishnoi likes to point out on his blog, an often overlooked source of wisdom is Willem Haemer’s PhD thesis from 1979. Many of Haemer’s proofs rely on simple properties of partitions of Hermitean matrices. My motivation for this post was a small exercise for myself. I wanted to prove the easy one of […]
At the moment I am very busy writing things like grant applications and research papers, so I lack the time for blog posts. But then I wasted part of my evening reading this article on FiveThiryEight about the Democratic Party primaries in the US. For each democratic primary contender, they provide the following data: How […]
I spent the last few days in vain using several spectral arguments to bound the size of certain intersection problems. For instance what is the largest set of vectors in $ {\{ 0, 1 \}^4}$ pairwise at Hamming distance at most $ {2}$ (a problem solved by Kleitman, recently investigated by Huang, Klurman and Pohoata). […]
Let us consider the $ {n}$-dimensional hypercube $ {\{ 0, 1 \}^n}$. The Hamming graph on $ {H_n}$ has the elements of $ {\{ 0, 1 \}^n}$ as vertices an two vertices are adjacent if their Hamming distance is one, so they differ in one coordinate. It is easy to see that the independence number […]
1. Introduction The following is mostly based on texts by Cordula Tollmien. I thank John Bamberg for his assistance, and Cordula Tollmien and Cheryl Praeger for their helpful comments on earlier drafts of this text. Emmy Noether is one the most influential mathematicians of all time and one of the shining examples of the mathematics […]