We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We consider relations between thresholds for monotone set properties and simple lower bounds for such thresholds. A motivating example (Conjecture 2): Given an n-vertex graph H, write pE for the least p such that, for each subgraph H' of H, the expected number of copies of H' in G=G(n, p) is at least 1, and pc for that p for which the probability that G contains (a copy of) H is 1/2. Then (conjecture) pc=O(pElog n). Possible connections with discrete isoperimetry are also discussed.

Thresholds and expectation thresholds

The classical Erdős-Szekeres theorem dating back almost a hundred years states that any sequence of (n − 1) + 1 distinct real numbers contains a monotone subsequence of length n. This theorem has been generalised to higher dimensions in a variety of ways but perhaps the most natural one was proposed by Fishburn and Graham more than 25 years ago. They defined the concept of a monotone and a lex-monotone array and asked how large an array one needs in order to be able to find a monotone or a lex-monotone subarray of size n× . . .×n. Fishburn and Graham obtained Ackerman-type bounds in both cases. We significantly improve these results. Regardless of the dimension we obtain at most a triple exponential bound in n in the monotone case and a quadruple exponential one in the lex-monotone case.

Erdős–Szekeres theorem for multidimensional arrays

In 1973, Erd˝os conjectured the existence of high girth ( n, 3 , 2)-Steiner systems. Recently, Glock, K¨uhn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erd˝os’ conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erd˝os’ conjecture. As for Steiner systems with more general parameters, Glock, K¨uhn, Lo, and Osthus conjectured the existence of high girth ( n, q, r )-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern ﬁnding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V ( H ) = E ( G )). Our ﬁrst main result is a common generalization of the classical theorems of Pippenger (for ﬁnding an almost perfect matching) and Ajtai, Koml´os, Pintz, Spencer, and Szemer´edi (for ﬁnding an independent set in girth ﬁve hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a speciﬁc instance). Indeed, the coloring version of our result even yields an almost partition of K rn into approximate high girth ( n, q, r )-Steiner systems. Our main results also imply the existence of a perfect matching in a bipartite hypergraph where the parts have slightly unbalanced degrees. This has a number of other applications; for example, it proves the existence of ∆ pairwise disjoint list colorings in the setting of Kahn’s theorem for list coloring the edges of a hypergraph; it also proves asymptotic versions of various rainbow matching results in the sparse setting (where the number of times a color appears could be much smaller than the number of colors) and even the existence of many pairwise disjoint rainbow matchings in such circumstances.

Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings

In this paper, we study the problem of computing a tight lower bound on the dimension of the strong structurally controllable subspace (SSCS) in networks with Laplacian dynamics. The bound is based on a sequence of vectors containing the distances between leaders (nodes with external inputs) and followers (remaining nodes) in the underlying network graph. Such vectors are referred to as the distance-to-leaders vectors. {We give exact and approximate algorithms to compute the longest sequences of distance-to-leaders vectors, which directly provide distance-based bounds on the dimension of SSCS. The distance-based bound is known to outperform the other known bounds (for instance, based on zero-forcing sets), especially when the network is partially strong structurally controllable. Using these results, we discuss an application of the distance-based bound in solving the leader selection problem for strong structural controllability. Further, we characterize strong structural controllability in path and cycle graphs with a given set of leader nodes using sequences of distance-to-leaders vectors. Finally, we numerically evaluate our results on various graphs.

Computation of the Distance-based Bound on Strong Structural Controllability in Networks

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erd\H{o}s-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of length $\Omega_{k}\left(\sqrt{n}\right)$, and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random $k$-dimensional permutation of order $n$ is asymptotically almost surely $\Theta_{k}\left(n^{\frac{k}{k+1}}\right)$.

Monotone Subsequences in High-Dimensional Permutations

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For every k ≥ 1, every order-n k-dimensional permutation contains a monotone subsequence of length Ω k ( ), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θ k (n k/(k+1)).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/239407E0538A5D9C3430A7D8F71F4B76/S0963548317000517a.pdf/div-class-title-monotone-subsequences-in-high-dimensional-permutations-div.pdf

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices. As long as the smallest degree $\delta (G)<k$, choose, uniformly at random, two vertices $u,v \in V(G)$ of degree $\delta(G)$ whose distance is at least $g-1$. If there are no such vertex pairs, abort. Otherwise, add the edge $uv$ to $E(G)$. 
We show that with high probability this algorithm yields a $k$-regular graph with girth at least $g$. Our analysis also implies that there are $\left( \Omega (n) \right)^{kn/2}$ labeled $k$-regular $n$-vertex graphs with girth at least $g$.

A randomized construction of high girth regular graphs

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary $k$-regular bipartite graphs $G$ on $2n$ vertices for all $k = \omega \left( \frac{n}{\log^{1/3} n} \right)$. 
Surprisingly, this is not the case for smaller values of $k$. Using a construction due to Goel, Kapralov and Khanna, we show that there exist bipartite $k$-regular graphs in which the last isolated vertex disappears long before a perfect matching appears.

Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs

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Papers

Monotone Subsequences in High-Dimensional Permutations

Monotone Subsequences in High-Dimensional Permutations

A randomized construction of high girth regular graphs

A randomized construction of high girth regular graphs

Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs