Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of equiangular lines with fixed angles. It led to the natural question: what is the maximum second eigenvalue multiplicity of a connected bounded degree 𝑛-vertex graph? The best-known upper bound is 𝑂(𝑛∕ log log 𝑛). The previously known bestknown lower bound is on the order of 𝑛 1∕3 (for infinitely many 𝑛), coming from Cayley graphs on PSL(2, 𝑞).Here we give a construction showing a lower bound of √ 𝑛∕ log 2 𝑛. We also construct Cayley graphs with second eigenvalue multiplicity at least 𝑛 2∕5 − 1. Earlier techniques show that there are at most 𝑂(𝑛∕ log log 𝑛) eigenvalues (counting multiplicities) within 𝑂(1∕ log 𝑛)of the second eigenvalue. We give a construction showing this upper bound on approximate second eigenvalue multiplicity is tight up to a constant factor. This demonstrates a barrier to earlier techniques for upper bounding eigenvalue multiplicities.

Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of equiangular lines with fixed angles. It led to the natural question: what is the maximum second eigenvalue multiplicity of a connected bounded degree n-vertex graph? The best known upper bound is O(n/ log log n). The previously known best known lower bound is on the order of n 1/3 (for infinitely many n), coming from Cayley graphs on PSL(2, q).Here we give constructions showing a lower bound on the order of n/ log n. We also construct Cayley graphs with second eigenvalue multiplicity at least n 2/5 − 1. Earlier techniques show that there are at most O(n/ log log n) eigenvalues (counting multiplicities) within O(1/ log n) of the second eigenvalue. We give a construction showing this upper bound on approximate second eigenvalue multiplicity is tight up to a constant factor. This demonstrates a barrier to earlier techniques for upper bounding eigenvalue multiplicities.

The Dushnik-Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza showed that the dimension of the divisibility order on the interval of integers [N/κ, N ] is bounded above by κ(log κ) 1+o(1) and below by Ω((log κ/ log log κ) 2 ). We improve the upper bound to O((log κ) 3 /(log log κ) 2 ). We also prove a more general result on posets of multisets ordered by inclusion.