We use a graph-theoretic approach which yields improvements on the known Gilbert-Varshamov (GV) bound for sum-rank-metric codes for certain parameters. In particular, we show that asymptotically Fn×mqcan be partitioned into sum-rank-metric codes whose average size is bigger than the GV bound by a logarithmic factor for these parameters. Finally, we discuss the connection of such codes to set-coloring Ramsey numbers.

Abstract. The famous Erdős–Sós conjecture states that every graph of average degree more than [Formula: see text] must contain every tree on [Formula: see text] vertices. In this paper, we study a spectral version of this conjecture. For [Formula: see text], let [Formula: see text] be the join of a clique on [Formula: see text] vertices with an independent set of [Formula: see text] vertices and denote by [Formula: see text] the graph obtained from [Formula: see text] by adding one edge. We show that for fixed [Formula: see text] and sufficiently large [Formula: see text], if a graph on [Formula: see text] vertices has adjacency spectral radius at least as large as [Formula: see text] and is not isomorphic to [Formula: see text], then it contains all trees on [Formula: see text] vertices. Similarly, if a sufficiently large graph has spectral radius at least as large as [Formula: see text], then it either contains all trees on [Formula: see text] vertices or is isomorphic to [Formula: see text]. This answers a two-part conjecture of Nikiforov affirmatively.

An $m$-general set in $AG(n,q)$ is a set of points such that any subset of size $m$ is in general position. A 3-general set is often called a capset. In this paper, we study the maximum size of an $m$-general set in $AG(n,q)$, significantly improving previous results. When $m=4$ and $q=2$, we give a precise estimate, solving a problem raised by Bennett.

In this paper we study the maximum number of hyperedges which may be in an $r$-uniform hypergraph under the restriction that no pair of vertices has more than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the even-cycle problem asking for ${ex}(n, C_{2k})$. We extend results of Füredi and Simonovits and of Conlon, who studied the problem when $r=2$. In particular, we show that for fixed $k$ and $r$, there is a constant $t$ such that the maximum number of edges can be determined in order of magnitude.

Given a projective plane $\Sigma$ and a polarity $\theta$ of $\Sigma$, the corresponding polarity graph is the graph whose vertices are the points of $\Sigma$, and two distinct points $p_1$ and $p_2$ are adjacent if $p_1$ is incident to $p_2^{ \theta}$ in $\Sigma$. A well-known example of a polarity graph is the Erdös--Rényi orthogonal polarity graph $ER_q$, which appears frequently in a variety of extremal problems. Eigenvalue methods provide an upper bound on the independence number of any polarity graph. Mubayi and Williford showed that in the case of $ER_q$, the eigenvalue method gives the correct upper bound in order of magnitude. We prove that this is also true for certain other families of polarity graphs. This includes a family of polarity graphs for which the polarity is neither orthogonal nor unitary. We conjecture that any polarity graph of a projective plane of order $q$ has an independent set of size $\Omega (q^{3/2})$. Some related results are also obtained.