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Minors in random and expanding hypergraphs
 in 27th SoCG’, ACM
, 2011
"... We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, ..."
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We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, we focus on threshold problems. The basic model for random complexes is the LinialMeshulam model X k (n, p). By definition, such a complex has n vertices, a complete (k − 1)dimensional skeleton, and every possible kdimensional simplex is chosen independently with probability p. We show that for every k, t ≥ 1, there is a constant C = C(k, t) such that for p ≥ C/n, the random complex X k (n, p) asymptotically almost surely contains K k t (the complete kdimensional complex on t vertices) as a homological minor. As corollary, the threshold for (topological) embeddability of X k (n, p) into R 2k is at p = Θ(1/n). The method can be extended to other models of random complexes (for which the lower skeleta are not necessarily complete) and also to more general Tverbergtype problems, where instead of continuous maps without doubly covered image points (embeddings), we consider maps without qfold covered image points.
Probability and Geometry on Groups  Lecture notes for a graduate course
, 2015
"... These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley grap ..."
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These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley graphs of infinite groups; 2) the algebraic properties of these groups; and 3) the behaviour of probabilistic processes (most importantly, random walks, harmonic functions, and percolation) on these Cayley graphs. I try to be as little abstract as possible, emphasizing examples rather than presenting theorems in their most general forms. I also try to provide guidance to recent research literature. In particular, there are presently over 150 exercises and many open problems that might be accessible to PhD students. It is also hoped that researchers working either in probability or in geometric group theory will find these notes useful to enter the other field.
On the nonplanarity of a random subgraph
, 2012
"... Let G be a finite graph with minimum degree r. Form a random subgraph Gp of G by taking each edge of G into Gp independently and with probability p. We prove that for any constant ǫ> 0, if p = 1+ǫ r, then Gp is nonplanar with probability approaching 1 as r grows. This generalizes classical resul ..."
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Let G be a finite graph with minimum degree r. Form a random subgraph Gp of G by taking each edge of G into Gp independently and with probability p. We prove that for any constant ǫ> 0, if p = 1+ǫ r, then Gp is nonplanar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs. 1
Small minors in dense graphs
 European J. Combin
"... Abstract. A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe funct ..."
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Abstract. A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe functions f and h such that every graph with n vertices and average degree at least f(t) contains a Ktmodel with at most h(t) · logn vertices. The logarithmic dependence on n is best possible (for fixed t). In general, we prove that f(t) ≤ 2t−1 + ε. For t ≤ 4, we determine the least value of f(t); in particular f(3) = 2 + ε and f(4) = 4 + ε. For t ≤ 4, we establish similar results for graphs embedded on surfaces, where the size of the Ktmodel is bounded (for fixed t). 1.