Random cliques in random graphs

We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed, in the (one-sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus, an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem -- recently announced by Jeff Kahn -- implies a corresponding bound for the threshold for $G(n,p)$ to contain a $K_r$-factor. We also prove a slightly weaker result for $r=3$, and (weaker) generalizations replacing $K_r$ by certain other graphs $F$. As an application of the latter we find, up to a log factor, the threshold for $G(n,p)$ to contain an $F$-factor when $F$ is $1$-balanced but not strictly $1$-balanced.Comment: 19 pages; expanded introduction and Section 5, plus minor correction

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices

A Unifying Hierarchy of Valuations with Complements and Substitutes

We introduce a new hierarchy over monotone set functions, that we refer to as $\mathcal{MPH}$ (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, $\mathcal{MPH}$-$m$ (where $m$ is the total number of items) captures all monotone functions. The lowest level, $\mathcal{MPH}$-$1$, captures all monotone submodular functions, and more generally, the class of functions known as $\mathcal{XOS}$. Every monotone function that has a positive hypergraph representation of rank $k$ (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in $\mathcal{MPH}$-$k$. Every monotone function that has supermodular degree $k$ (in the sense defined by Feige and Izsak [ITCS 2013]) is in $\mathcal{MPH}$-$(k+1)$. In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of $\mathcal{MPH}$-$k$. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the $\mathcal{MPH}$ hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of $k+1$ if all players hold valuation functions in $\mathcal{MPH}$-$k$. The other is an upper bound of $2k$ on the price of anarchy of simultaneous first price auctions. Being in $\mathcal{MPH}$-$k$ can be shown to involve two requirements -- one is monotonicity and the other is a certain requirement that we refer to as $\mathcal{PLE}$ (Positive Lower Envelope). Removing the monotonicity requirement, one obtains the $\mathcal{PLE}$ hierarchy over all non-negative set functions (whether monotone or not), which can be fertile ground for further research

Percolation on self-dual polygon configurations

Recently, Scullard and Ziff noticed that a broad class of planar percolation models are self-dual under a simple condition that, in a parametrized version of such a model, reduces to a single equation. They state that the solution of the resulting equation gives the critical point. However, just as in the classical case of bond percolation on the square lattice, self-duality is simply the starting point: the mathematical difficulty is precisely showing that self-duality implies criticality. Here we do so for a generalization of the models considered by Scullard and Ziff. In these models, the states of the bonds need not be independent; furthermore, increasing events need not be positively correlated, so new techniques are needed in the analysis. The main new ingredients are a generalization of Harris's Lemma to products of partially ordered sets, and a new proof of a type of Russo-Seymour-Welsh Lemma with minimal symmetry assumptions.Comment: Expanded; 73 pages, 24 figure