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

The Theory of Information and Coding

A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all of them. Erdős and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower with $r$ petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to $c^w$ for some constant $c$. In this paper, we improve the bound to about $(\log w)^w$. In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is sharp up to lower order terms.

Improved bounds for the sunflower lemma

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

Coding for sunflowers, Discrete Analysis 2020:2, 8 pp.

The sunflower problem is an old problem of Erdős and Rado in extremal set theory. A _sunflower_ is defined to be a collection of sets $A_1,\dots,A_r$ such that if $B=\bigcap_{i=1}^rA_i$ is their intersection, then the sets $A_i\setminus B$ are disjoint. Equivalently, all the pairwise intersections $A_i\cap A_j$ (with $i\ne j$) are the same. Erdős asked how many sets of size $k$ there can be without a sunflower of size $r$. To see that there is some finite bound, observe that if one starts with $m$ sets, then either one can find $r$ of them that are disjoint, in which case one has a sunflower, or one can find a set $A$ that intersects at least $m/r-1$ of the other sets. By the pigeonhole principle, there is an element of $A$ that is contained in at least $k^{-1}(m/r-1)$ of the other sets. Removing that element from each of those sets gives a collection of sets of size $k-1$ and one can apply induction.

A suitably careful version of this argument gives an upper bound on $m$ of $k!(r-1)^{k}$. Erdős and Rado conjectured that the correct bound is exponential in $k$ for fixed $r$. This problem is still open even for $r=3$.

Until recently, the best known bound did not improve much on this factorial bound. The record when $r=3$, due to Kostochka, was $Ck!(\log\log\log k/\log\log k)^k$. To get a feel for this, note that $k!$ is approximately $(k/e)^k$, so Kostochka replaced the linear function $k/e$ by a function that was sublinear, but only because of a $\log\log$ factor. Then in 2019, Alweiss, Lovett, Wu and Zhang improved the bound dramatically, to one of the form $(\log k)^k(r\log\log k)^{O(k)}$, replacing Kostochka's almost linear function by a logarithmic one. If one imagines a spectrum of growth rates with exponential at one end and factorial at the other, this replaced a growth rate that was almost at the factorial end by one that was almost at the exponential end. 

To achieve this, they obtained an upper bound for the number of sets of size $k$ one could find without a structure called a _robust sunflower_, which is a collection of sets $A_1,\dots,A_r$ with intersection $B$ such that if elements of $\bigcup_i(A_i\setminus B)$ are chosen independently at random with probability $\alpha$, then with probability at least $1-\beta$ there is some $i$ such that all the elements of $A_i\setminus B$ are chosen. It can be shown that if $\alpha=\beta=r^{-1}$, then a robust sunflower with parameters $\alpha$ and $\beta$ contains a sunflower of size $r$. Interestingly, their upper bound for set systems without robust sunflowers was close to sharp, so they reached the natural barrier for their method. A subsequent paper of Frankston, Kahn, Narayanan and Park removed the $\log\log k$ factor from the bound, giving a bound of $(C\log k\log(rk))^k$.

The main purpose of this paper is to give a shorter and cleaner proof of the result of Alweiss, Lovett, Wu and Zhang, using the converse of Shannon's noiseless coding theorem. In fact, it achieves the same bound as Frankston, Kahn, Narayanan and Park, and also gives essentially sharp estimates for the size of a set system without a robust sunflower with parameters $\alpha$ and $\beta$, which have other applications in theoretical computer science.

/pdf/coding-for-sunflowers-1wtpm3f7xk.pdf

Coding for Sunflowers

Proving a conjecture of Talagrand, a fractional version of the 'expectation-threshold' conjecture of Kalai and the second author, we show for any increasing family $F$ on a finite set $X$ that $p_c (F) =O( q_f (F) \log \ell(F))$, where $p_c(F)$ and $q_f(F)$ are the threshold and 'fractional expectation-threshold' of $F$, and $\ell(F)$ is the largest size of a minimal member of $F$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson--Kahn--Vu), bounded-degree spanning trees (Montgomery), and bounded-degree spanning graphs (new). We also resolve (and vastly extend) the 'axial' version of the random multi-dimensional assignment problem (earlier considered by Martin--Mezard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdős--Rado 'Sunflower Conjecture'.

Thresholds versus fractional expectation-thresholds

Ellis and the third author showed, verifying a conjecture of Frankl, that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ admitting a transitive automorphism group has cardinality $o(2^n)$, while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ that is regular and increasing has cardinality $o(2^n)$.

On regular 3-wise intersecting families

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional Expectation-Threshold Conjecture," due to Talagrand and recently proved by the authors and B. Narayanan, and the (stronger) "Expectation-Threshold Conjecture" of the second author and G. Kalai. The conjecture under discussion here says there is a fixed $L$ such that if, for a given $\mathcal F$, $p\in [0,1]$ admits $\lambda:2^V \rightarrow \mathbb R^+$ with \[ \mbox{$\sum_{S\subseteq F}\lambda_S\ge 1 ~~\forall F\in \mathcal F$} \] and \[ \mbox{$\sum_S\lambda_Sp^{|S|} \le 1/2$} \] (a.k.a. $\mathcal F$ is weakly $p$-small), then $p/L$ admits such a $\lambda$ taking values in $\{0,1\}$ ($\mathcal F$ is $(p/L)$-small). Talagrand showed this when $\lambda$ is supported on singletons and suggested, as a more challenging test case, proving it when $\lambda$ is supported on pairs. The present work provides such a proof.

Keith Frankston

Papers

Thresholds versus fractional expectation-thresholds

Thresholds versus fractional expectation-thresholds

On regular 3-wise intersecting families

On regular 3-wise intersecting families

On a problem of M. Talagrand