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Friday, November 7, 2014

**Abstract:** First I will define the stability function $g_T(\kappa):=\sup_{M\models T, |M|=\kappa}|S^1(M)|$ for a complete theory $T$ (where $S^1(M)$ is the Stone Space of 1-types with parameters from $M$). Next, I will recall some dividing lines associated to $g_T$. Then, I will give a proof that for a NIP (=Not the Independence Property) theory $T$, we have $g_T(\kappa)\leq ded(\kappa)^{\aleph_0}$, whereas $g_T(\kappa) = 2^{\kappa}$ in the IP (not NIP) case. ($ded(\kappa)$ is roughly the maximum number of Dedekind cuts a linear order of size $\kappa$ can have). Finally, I will mention how some set-theoretic black magic shows that NIPness can always be detected from the function $g_T(\kappa)$. I will use this last thing in my proof of NIP for $T_{\log}$ in a future talk.