33 research outputs found

    On the lower tail variational problem for random graphs

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    We study the lower tail large deviation problem for subgraph counts in a random graph. Let XHX_H denote the number of copies of HH in an Erd\H{o}s-R\'enyi random graph G(n,p)\mathcal{G}(n,p). We are interested in estimating the lower tail probability P(XH≤(1−δ)EXH)\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H) for fixed 0<δ<10 < \delta < 1. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p≥n−αHp \ge n^{-\alpha_H} (and conjecturally for a larger range of pp). We study this variational problem and provide a partial characterization of the so-called "replica symmetric" phase. Informally, our main result says that for every HH, and 0<δ<δH0 < \delta < \delta_H for some δH>0\delta_H > 0, as p→0p \to 0 slowly, the main contribution to the lower tail probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite HH and δ\delta close to 1.Comment: 15 pages, 5 figures, 1 tabl

    Combinatorics

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    Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

    Bounds on Ramsey Games via Alterations

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    This note contains a refined alteration approach for constructing H-free graphs: we show that removing all edges in H-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdos and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers.Comment: 9 page

    Counting extensions revisited

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    We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.Comment: 21 pages, 2 figure
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