Igor Balla

Let $x_1, \ldots, x_n \in \mathbb{R}^d$ be unit vectors such that among any three there is an orthogonal pair. How large can $n$ be as a function of $d$, and how large can the length of $x_1 + \ldots + x_n$ be? The answers to these two celebrated questions, asked by Erd\H{o}s and Lov\'{a}sz, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lov\'{a}sz $\vartheta$-function and minimum semidefinite rank. In this paper, we study these parameters for general $H$-free graphs. In particular, we show that for certain bipartite graphs $H$, there is a connection between the Tur\'{a}n number of $H$ and the maximum of $\vartheta \left( \overline{G} \right)$ over all $H$-free graphs $G$.

The minrank of a graph $G$ on the set of vertices $[n]$ over a field $\mathbb{F}$ is the minimum possible rank of a matrix $M\in\mathbb{F}^{n\times n}$ with nonzero diagonal entries such that $M_{i,j}=0$ whenever $i$ and $j$ are distinct nonadjacent vertices of $G$. This notion, over the real field, arises in the study of the Lov\'asz theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph $G(n,p)$ over any finite or infinite field, showing that for every field $\mathbb{F}=\mathbb F(n)$ and every $p=p(n)$ satisfying $n^{-1} \leq p \leq 1-n^{-0.99}$, the minrank of $G=G(n,p)$ over $\mathbb{F}$ is $\Theta(\frac{n \log (1/p)}{\log n})$ with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev, and Weinstein, who proved the above result for finite fields of size at most $n^{O(1)}$, with tools from linear algebra, including an estimate of R\'onyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials.

A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in $\mathbb{R}^n$ was studied extensively for the last 70 years. In this paper, we study analogous questions for $k$-dimensional subspaces. We discuss natural ways of defining the angle between $k$-dimensional subspaces and correspondingly study the maximum size of an equiangular set of $k$-dimensional subspaces in $\mathbb{R}^n$. Our bounds extend and improve a result of Blokhuis.

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $\theta$. Moreover, this is achievable only for $\theta = \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k)$ lines in $\mathbb{R}^n$ having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.

Akbari, Etesami, Mahini, and Mahmoody conjectured that every proper edge colouring of $K_n$ with $n$ colours contains a Hamilton cycle with $\leq O(\log n)$ colours. They proved that there is always a Hamilton cycle with $\leq 8\sqrt n$ colours. In this note we improve this bound to $O(\log^3 n)$.

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(\chi(H)-1)+\sigma(H)$, where $\chi(H)$ is the chromatic number of $H$ and $\sigma(H)$ is the size of the smallest color class in a $\chi(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(\chi(H)-1)+\sigma(H)$. The notion of Ramsey goodness was introduced by Burr and Erd\H{o}s in 1983 and has been extensively studied since then. In this paper we show that if $n\geq \Omega(|H| \log^4 |H|)$ then every $n$-vertex bounded degree tree $T$ is $H$-good. The dependency between $n$ and $|H|$ is tight up to $\log$ factors. This substantially improves a result of Erd\H{o}s, Faudree, Rousseau, and Schelp from 1985, who proved that $n$-vertex bounded degree trees are $H$-good when when $n \geq \Omega(|H|^4)$.