inertia bound

The Independence Number of the Orthogonality Graph — Or: The Usefulness of Literature Study

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 many publications which investigate this problem. The aim of this post is two fold:

  1. To summarize the state of the art.
  2. To demonstrate how careful literature study is helpful to obtain results.


Six Spectral Bounds

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). Here the answer is {5} and {0000}, {0001}, {0010}, {0100}, {1000} would be such an example. Or the largest set of vectors in {\{ 0, 1 \}^7} pairwise at distance {2} or {6}. Here the answer is {8} and an example is {0000001}, {0000010}, {0000100}, {0001000}, {0010000}, {0100000}, {1000000}, {1111111}. This is a problem which I recently investigated together with Hajime Tanaka, extending work by Frankl and others.

Usually, this playing around does not lead to anything. But this time …. It is actually the same. However, I did one useful thing which is the following: Generously counting, I do know five different easy spectral arguments which can be used to investigate these questions. This blog post presents these methods for the two problems mentioned above.

Huang’s Breakthrough, Cvetković’s Bound, Godsil’s Question, and Sinkovic’s Answer

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 {\alpha} of this graph is {2^{n-1}}.

It was a long open and famous problem what the maximum degree of an induced subgraph on {H_n} with {\alpha+1} vertices is. Very recently, Hao Huang showed that the answer is “at least {\sqrt{n}}” and everyone is blogging about it (only a small selection): Anurag Bishnoi, Gil Kalai, Terry Tao, Fedya Petrov. Here I am jumping on this bandwagon.

Huang uses a variant of the inertia bound (or Cvetković bound). It is a good friend of the ratio bound (or Hoffman’s bound) which is the namesake of this blog. For the second time this year (the first time was due to a discussion with Aida Abiad), this I was reminded me of a result by John Sinkovic from 3 years ago. This blog posts is about Sinkovic’s result which answered a question by Chris Godsil on the inertia bound.