Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1807.06249

Equiangular lines and the Lemmens-Seidel conjecture

Yen-chi Roger Lin·Wei-Hsuan Yu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle $1/5$ are proved by carefully analyzing pillar decompositions, with the aid of the uniqueness of two-graphs on $276$ vertices. The Neumann Theorem is generalized in the sense that if there are more than $2r-2$ equiangular lines in $\mathbb{R}^r$, then the angle is quite restricted. Together with techniques on finding saturated equiangular sets, we determine the maximum size of equiangular sets "exactly" in an $r$-dimensional Euclidean space for $r = 8$, $9$, and $10$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yen-chi Roger Lin, Wei-Hsuan Yu. 2019-08-29. Equiangular lines and the Lemmens-Seidel conjecture. https://arxiv.org/abs/1807.06249

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maximal number of mixed Nash equilibria in generic games where each player has two pure strategies

The number of Nash equilibria of the mixed extension of a generic finite game in normal form is finite and odd. This raises the question how large the number can be, depending on the number of players and the numbers of their pure strategies. Here we present a lower bound for the maximal possible number in the case of m-player games where each player has two pure strategies. It is surprisingly close to a known upper bound.

math.CO↗
arXiv2026

Simultaneous avoidance of length-4 patterns in ascent sequences

Ascent sequences form a central class of combinatorial objects, as they are in bijection with several important families such as (2+2)-free posets, Stoimenow matchings, and other Fishburn objects, and are enumerated by the Fishburn numbers. We study pattern avoidance in ascent sequences for the five patterns of length 4: $0101$, $0102$, $0112$, $0120$, and $0121$. These patterns arise naturally from recent work on pattern avoidance in related families of Fishburn objects, including Stoimenow matchings and (2+2)-free posets. We enumerate ascent sequences avoiding any subset of these patterns, with the exception of the sets $\{0120\}$, $\{0121\}$, and $\{0120,0121\}$, for which the enumeration remains open. Our results reveal that the corresponding avoidance classes fall into $16$ Wilf equivalence classes and exhibit a wide range of enumerative behaviour, including connections to classical sequences such as the Catalan and Fibonacci numbers, as well as polynomial formulas and rational generating functions; several of the sequences we obtain appear to be new. Our methods combine structural decompositions with generating-tree techniques and, in several cases, rely on reductions to shorter patterns via restricted growth functions. This work contributes to the broader study of pattern avoidance across Fishburn families and highlights further connections between ascent sequences and other combinatorial structures.

math.CO↗