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For each n, let Xn ∈ {0, . . . , n} be a random variable with mean µn, standard deviation σn, and let Pn(z) = n k=0 P(Xn = k)z k , be its probability generating function. We show that if none of the complex zeros of the polynomials {Pn(z)} is contained in a neighbourhood of 1 ∈ C and σn > n ε for some ε > 0, then X * n = (Xn − µn)σ −1 n is asymptotically normal as n → ∞: that is tends in distribution to a random variable Z ∼ N (0, 1).Moreover, we show this result is sharp in the sense that there exist sequences of random variables {Xn} with σn > C log n for which Pn(z) has no roots near 1 and X * n is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of Pn(z) and the distribution of the random variable Xn. 1 arXiv:1804.07696v2 [math.PR] 12 Jun 2018 P X (z) are non-negative. Hence, we may write P X (z) = n i=1 (q i z + 1 − q i ), for some q 1 , . . . q n ∈ [0, 1]. A little further thought reveals that this special expression for the probability generating function corresponds to an expression of X as a sum of independent random variables X = X 1 + · · · + X n , where X i is the {0, 1}-random variable, taking 1 with probability q i . Thus, with an appropriate central limit theorem at hand, we see that X must be approximately normal, provided the variance of X is sufficiently large. In other words, from this one piece of information (albeit a strong piece of information) about the zeros of P X (z), one can quickly deduce quite a bit of information about the distribution of X. The aim of this paper is to show that this assumption of real rootedness of P X (z) can be related quite considerably while yielding similar results. In particular, we give three different results each of which says that "if X has large variance and the roots of P X (z) avoid a region in the complex plane, then X is approximately normal." HistoryBefore turning to our contributions, we take a brief moment to situate our results in an old and well-studied field centred around the following question: What does information about the coefficients of P (z) tell us about the distribution of the complex roots of P (and vice versa)? This question has a long and rich history, reaching back to the seminal work of Littlewood, Szegő, Pólya, and perhaps even Cauchy, due to his 1829 proof [8] of the fundamental theorem of algebra, which gives explicit bounds on the magnitude of the complex roots (see [7, Theorem 1.2.1] for a modern treatment of this proof). One line of research, initiated by the 1938 -1943 work of Littlewood and Offord [21, 22, 23], concerns the typical distribution of roots of random polynomials. For example Kac [18] gave an exact integral formulafor the number of real roots of random polynomial, with coefficients sampled independently from a normal distribution. Later, Erdős and Offord [11] showed that as n → ∞ almost all polynomials of the form n i=0 ε i x i , where ε 1 , . . . , ε n ∈ {0...

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For r ≥ 2, we show that every maximal K r+1 -free graph G on n vertices with (1 − 1 r ) n 2 2 − o(n r+1 r ) edges contains a complete r-partite subgraph on (1 − o(1))n vertices. We also show that this is best possible. This result answers a question of Tyomkyn and Uzzell.The classical theorem of Turán [12] tells us that, for an integer r ≥ 2, the maximum number of edges in a graph not containing a K r+1 is t r (n), and that T r (n) is the unique K r+1 -free graph attaining this maximum. Erdős and Simonovits [6,5,11] discovered that this extremal problem exhibits a certain 'stability' phenomenon: K r+1 -free graphs for which e(G) is close to t r (n) must resemble the Turán graph in an appropriate sense. In particular, they proved that every n-vertex, K r+1 -free graph with at least t r (n) − o(n 2 ) edges can be transformed into T r (n) by making at most o(n 2 ) edge deletions and additions.Beyond the seminal work of Erdős and Simonovits, we are lead to consider finer aspects of this phenomenon. More generally, it is natural to ask how the structure of a K r+1 -free graph G comes to resemble the Turán graph as the number of edges e(G) approaches the Turán number t r (n). For instance, Nikiforov and Rousseau [10], in the context of a Ramsey-theoretic problem, showed that for r ≥ 2 and ε sufficiently small (depending on r) the following holds:if G is an n-vertex K r+1 -free graph with e(G) ≥ 1 − 1 r − ε n 2 /2, then G contains an induced r-partite subgraph H with |H| ≥ (1 − 2ε 1/3 )n and δ(H) ≥ 1 − 1 r − 4ε 1/3 n. In other words, G must contain a large r-partite subgraph with minimum degree almost as large as δ(T r (n)).The interested reader should consult the survey of Nikiforov [9] for a few other stability results in a similar vein.

Abstract. A graph on n vertices is said to be C-Ramsey if every clique or independent set of the graph has size at most C log n. The only known constructions of Ramsey graphs are probabilistic in nature, and it is generally believed that such graphs possess many of the same properties as dense random graphs. Here, we demonstrate one such property: for any fixed C > 0, every C-Ramsey graph on n vertices induces subgraphs of at least n 2−o(1) distinct sizes. This near-optimal result is closely related to two unresolved conjectures, the first due to Erdős and McKay and the second due to Erdős, Faudree and Sós, both from 1992.