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Razborov's plain flag algebra mehod has had great success on density problems for which the extremal example
is a blow-up of a small graph, i.e. every vertex is replaced by a clique or by a stable set. The method usually fails to deliver
sharp results when the extremal structure is an iterated
blow up, i.e. every vertex of a small graph is replaced by the graph itself repeatedly.
In this talk I will present results for the maximal induced densities of \(\vec{P_3}\), \(\vec{C_4}\), \(\vec{C_5}\), and \(C_5\).
In each case, the extremal structure is an iterated blow-up (of \(\vec{C_4}\), \(\vec{C_4}\), \(\vec{C_5}\), and \(C_5\) respectively),
and the plain flag algebra method fails to give sharp results. We use stability type results to show exact results, using the bounds from the flag algebra
calculations.
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