We consider Erdős-Ko-Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect nontrivially. We characterize the Erdős-Ko-Rado sets of generators of maximum size in all polar spaces, except for H(4n + 1, q 2 ) with n 2.

Abstract. We show that there are graphs with n vertices containing no K 5,5 which have about 1 2 n 7/4 edges, thus proving that ex(n, K 5,5 ) ≥ 1 2 (1 + o(1))n 7/4 . This bound gives an asymptotic improvement to the known lower bounds on ex(n, K t,s ) for t = 5 when 5 ≤ s ≤ 12, and t = 6 when 6 ≤ s ≤ 8.

In this paper, we investigate the minimum distance and small weight codewords of the LDPC codes of linear representations, using only geometrical methods. First we present a new lower bound on the minimum distance and we present a number of cases in which this lower bound is sharp. Then we take a closer look at the cases T * 2 (Θ) and T * 2 (Θ) D with Θ a hyperoval, hence q even, and characterize codewords of small weight. When investigating the small weight codewords of T * 2 (Θ) D , we deal with the case of Θ a regular hyperoval, i.e. a conic and its nucleus, separately, since in this case, we have a larger upper bound on the weight for which the results are valid.

A construction of Alon and Krivelevich gives highly pseudorandom K k -free graphs on n vertices with edge density equal to Θ(n −1/(k−2) ). In this short note we improve their result by constructing an infinite family of highly pseudorandom K k -free graphs with a higher edge density of Θ(n