On the Lovász δ-number of almost regular graphs with application to Erdős-Rényi graphs
European Journal of Combinatorics
Note: Tight embeddings of partial quadrilateral packings
Journal of Combinatorial Theory Series A
Some constructive bounds on Ramsey numbers
Journal of Combinatorial Theory Series B
Note: A new lower bound on the independence number of graphs
Discrete Applied Mathematics
The independence number for polarity graphs of even order planes
Journal of Algebraic Combinatorics: An International Journal
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The Erdős-Rényi and Projective Norm graphs arealgebraically defined graphs that have proved useful in supplyingconstructions in extremal graph theory and Ramsey theory. Theireigenvalues have been computed and this yields an upper bound ontheir independence number. Here we show that in many cases, thisupper bound is sharp in the order of magnitude. Our result for theErdős-Rényi graph has the following reformulation: themaximum size of a family of mutually non-orthogonal lines in avector space of dimension three over the finite field of orderq is of order q3/2. We also prove thatevery subset of vertices of size greater thanq2/2 + q3/2 +O(q) in the Erdős-Rényi graph contains atriangle. This shows that an old construction of Parsons isasymptotically sharp. Several related results and open problems areprovided. © 2007 Wiley Periodicals, Inc. J Graph Theory 56:113127, 2007