TL;DR
This paper investigates the properties of projective norm graphs, demonstrating that they contain certain complete bipartite subgraphs for infinitely many primes, which limits their extremal graph capabilities.
Contribution
It proves that P(q, 4) contains K_4,6 for infinitely many primes q, showing the limitations of these graphs in avoiding larger complete bipartite subgraphs.
Findings
P(q, 4) contains K_4,6 for infinitely many primes q
Limits the extremal properties of projective norm graphs
Shows these graphs cannot avoid certain bipartite subgraphs
Abstract
The projective norm graphs P(q, 4) introduced by Alon, R\'onyai and Szab\'o are explicit examples of extremal graphs not containing K_4,7. Ball and Pepe showed that P(q, 4) does not contain a copy of K_5,5 either for q >= 7, asymptotically improving the best lower bound for ex(n, K_5,5). We show that these results can not be improved, in the sense that P(q, 4) contains a copy of K_4,6 for infinitely many primes q.
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