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Jeff Kahn

Publications and source records attributed to Jeff Kahn.

At least 19 recordsLinked to original sources
arXiv2025

Variance vs. range for linear extensions, and balancing extensions in posets of bounded width

An old conjecture of Kahn and Saks says, roughly, that any poset $P$ of large enough width contains elements $x,y$ which are "balanced" in the sense that the probability that $x$ precedes $y$ in a uniformly random linear extension of $P$ is close to $1/2$. We show this implies the seemingly stronger statement that the same conclusion holds if, instead of large width, we assume only that, for some $x$, the number, $\pi(x)$, of elements of $P$ incomparable to $x$ is large. The implication follows from our two main results: first, that if $\pi(P):=\max \pi(x)$ is large then $P$ has large variance, i.e. there is a $y$ whose position in a uniform extension of $P$ has large variance; and second, that the conclusion of the Kahn-Saks Conjecture holds for $P$ with large variance and bounded width. These two assertions also yield an easy proof of a (not easy) result of Chan, Pak and Panova on "sorting probabilities" for Young diagrams, together with its natural generalization to higher dimensions.

math.CO↗
arXiv2025

On the "second" Kahn--Kalai Conjecture: cliques, cycles, and trees

We prove a few simple cases of a random graph statement that would imply the "second" Kahn--Kalai Conjecture. Even these cases turn out to be reasonably challenging, and it is hoped that the ideas introduced here may lead to further interest in, and further progress on, this natural problem.

math.CO↗
arXiv2025

Balancing Extensions in Posets of Large Width

We revisit classic balancing problems for linear extensions of a partially ordered set $P$, proving results that go far beyond many of the best earlier results on this topic. For example, with $p(x\prec y)$ the probability that $x$ precedes $y$ in a uniform linear extension, $\delta_{xy} = \min\{p(x \prec y), p(y \prec x)\}$, and $\delta(P)=\max \delta_{xy}$, we show that $\delta(P)$ tends to $1/2$ as $n := |P| \to\infty$ if $P$ has width $\Omega(n)$ or $\omega(\log(n))$ minimal elements, and is at least $1/e-o(1)$ if $P$ has width $\omega(\sqrt{n})$ or height $o(n)$. Motivated by both consequences for balance problems and intrinsic interest, we also consider several old and new parameters associated with $P$. Here, in addition to balance, we study relations between the parameters and suggest various questions that are thought to be worthy of further investigation.

math.CO↗
arXiv2025

On the "second" Kahn--Kalai Conjecture

We make progress on a conjecture of Kahn and Kalai, the original (stronger but less general) version of what became known as the ``Kahn-Kalai Conjecture" (KKC; now a theorem of Park and Pham). This ``second" KKC concerns the threshold, $p_c(H)$, for $G_{n,p}$ to contain a copy of a given graph $H$, predicting $p_c(H) = O(p_{\mathbb E}(H)\log n)$, where $p_{\mathbb E}$ is an easy lower bound on $p_c$. What we actually show is $p_{\mathbb E}^*(H)=O(p_{\mathbb E}(H)\log ^2n)$, where $p_{\mathbb E}^*$, the fractional expectation threshold, is a larger lower bound suggested by Talagrand. When combined with Talagrand's fractional relaxation of the KKC (now a theorem of Frankston, Kahn, Narayanan and Park), this gives $p_c(H)=O(p_{\mathbb E}(H)\log^3 n)$. (The second KKC would follow similarly if one could remove the log factors from the above bound on $p_{\mathbb E}^*$.)

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arXiv2024

Note on a conjecture of Talagrand: expectation thresholds vs. fractional expectation thresholds

We show that a restricted version of a conjecture of M. Talagrand on the relation between "expectation thresholds" and "fractional expectation thresholds" follows easily from a strong version of a second conjecture of Talagrand, on "selector processes." The selector process conjecture was proved by Park and Pham, and the quantitative strengthening used here is due to Bednorz, Martynek, and Meller.

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arXiv2024

On the $H$-space of a random graph

The edge space $\mathcal{E}(G)$ of a graph $G$ is the vector space $\mathbb{F}_2^{E(G)}$ with members naturally identified with subgraphs of $G$, and the $H$-space is the subspace $\mathcal{C}_H(G)$ of $ \mathcal{E}(G)$ spanned by copies of the graph $H$. We are interested in when the random graph $G = G_{n,p}$ is likely to satisfy \[\mathcal{C}_H(G) = \mathcal{W}_H(G),\] where $\mathcal{W}_H(G)$ takes one of four natural values, depending on the value of $\mathcal{C}_H(K_n)$. We show that for strictly $2$-balanced $H$, w.h.p. the above equality holds whenever every edge of $G$ is in a copy of $H$.

math.CO↗
arXiv2024

Asymptotics for Palette Sparsification from Variable Lists

It is shown that the following holds for each $\varepsilon >0$. For $G$ an $n$-vertex graph of maximum degree $D$, lists $S_v$ of size $D+1$ (for $v\in V(G)$), and $L_v$ chosen uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $S_v$ (independent of other choices), \[ \mbox{$G$ admits a proper coloring $\sigma$ with $\sigma_v\in L_v$ $\forall v$} \] with probability tending to 1 as $D\to \infty$. When each $S_v $ is $\{1\dots D+1\}$, this is an asymptotically optimal version of the ``palette sparsification'' theorem of Assadi, Chen and Khanna that was proved in an earlier paper by the present authors.

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arXiv2023

Asymptotics for Palette Sparsification

It is shown that the following holds for each $\varepsilon>0$. For $G$ an $n$-vertex graph of maximum degree $D$ and "lists" $L_v$ ($v \in V(G)$) chosen independently and uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $\{1, ..., D+1\}$, \[ G \text{ admits a proper coloring } \sigma \text{ with } \sigma_v \in L_v \forall v \] with probability tending to 1 as $D \to \infty$. This is an asymptotically optimal version of a recent "palette sparsification" theorem of Assadi, Chen, and Khanna.

math.CO↗
arXiv2022

A note on positive association

We show that if ${\mathcal A},{\mathcal B},{\mathcal C}$ are increasing subsets of $\Omega:=\{0,1\}^n$ with ${\mathcal A}\neq\emptyset$, then with respect to any product probability measure on $\Omega$, \[ \mbox{if each of the pairs $\{{\mathcal A}\cap{\mathcal B},{\mathcal C}\}$, $\{{\mathcal A}\cap {\mathcal C},{\mathcal A}\}$ is independent, then ${\mathcal B}$ and ${\mathcal C}$ are independent.} \] This implies an answer to a motivating question of J. Steif, and is related to a basic, still open variant of that question, and to a well-known conjecture of S. Sahi.

math.PR↗
arXiv2021

Linear cover time is exponentially unlikely

Proving a 2009 conjecture of Itai Benjamini, we show: For any C there is an $\varepsilon>0$ such that for any simple graph $G$ on $V$ of size $n$, and $X_0,\ldots$ an ordinary random walk on $G$, $P(\{X_0,\dots, X_{Cn}\}= V) < e^{-\varepsilon n}.$ A first ingredient in the proof of this is a similar statement for Markov chains in which all transition probabilities are sufficiently small relative to $C$.

math.PR↗
arXiv2021

On a problem of M. Talagrand

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.

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arXiv2020

Hitting times for Shamir's Problem

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the $M$-subsets of ${\mathcal K}:={V \choose r}$ ($:= \{\mbox{$r$-subsets of $V$}\}$). Shamir's Problem (circa 1980) asks, roughly, for what $M=M(n)$ is ${\mathcal H}$ likely to contain a perfect matching (that is, $n/r$ disjoint $r$-sets)? In 2008 Johansson, Vu and the author showed that this is true for $M>C_rn\log n$. More recently the author proved the asymptotically correct version of that result: for fixed $C> 1/r$ and $M> Cn\log n$, $P({\mathcal H} ~\mbox{contains a perfect matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}.$ The present work completes a proof, begun in that recent paper, of the definitive "hitting time" statement: $\mbox{Theorem.}$ If $A_1, \ldots ~$ is a uniform permutation of ${\mathcal K}$, ${\mathcal H}_t=\{A_1\dots A_t\}$, and \[ T=\min\{t:A_1\cup \cdots\cup A_t=V\}, \] then $P({\mathcal H}_T ~\mbox{contains a perfect matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}$.

math.CO↗
arXiv2020

Tuza's Conjecture for random graphs

A celebrated conjecture of Zs. Tuza says that in any (finite) graph, the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. Resolving a recent question of Bennett, Dudek, and Zerbib, we show that this is true for random graphs; more precisely: \[ \mbox{for any $p=p(n)$, $\mathbb P(\mbox{$G_{n,p}$ satisfies Tuza's Conjecture})\rightarrow 1 $ (as $n\rightarrow\infty$).} \]

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arXiv2019

Thresholds versus fractional expectation-thresholds

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--M\'{e}zard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erd\H{o}s--Rado 'Sunflower Conjecture'.

math.CO↗
arXiv2019

On symmetric intersecting families of vectors

A family of vectors $A \subset [k]^n$ is said to be intersecting if any two elements of $A$ agree on at least one coordinate. We prove, for fixed $k \ge 3$, that the size of a symmetric intersecting subfamily of $[k]^n$ is $o(k^n)$, which is in stark contrast to the case of the Boolean hypercube (where $k =2$). Our main contribution addresses limitations of existing technology: while there is now some spectral machinery, developed by Ellis and the third author, to tackle extremal problems in set theory involving symmetry, this machinery relies crucially on the interplay between up-sets and biased product measures on the Boolean hypercube, features that are notably absent in the problem at hand; here, we describe a method for circumventing these barriers.

math.CO↗
arXiv2019

Asymptotics for Shamir's Problem

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the $M$-subsets of ${\mathcal K}:={V \choose r}$ ($:= \{\mbox{$r$-subsets of $V$}\}$). Shamir's Problem (circa 1980) asks, roughly, for what $M=M(n)$ is ${\mathcal H}$ likely to contain a perfect matching (that is, $n/r$ disjoint $r$-sets)? In 2008 Johansson, Vu and the author showed that this is true for $M>C_rn\log n$. The present paper has two purposes. First, it establishes the asymptotically correct version of the 2008 result: Theorem 1. For fixed $\epsilon>0$ and $M> (1+\epsilon)(n/r)\log n$, $P({\mathcal H} ~\mbox{contains a perfect matching})\rightarrow 1 $ as $n\rightarrow\infty$. Second, it begins a proof of the definitive ``hitting time" statement: Theorem 2. If $A_1, \ldots ~$ is a uniform permutation of ${\mathcal K}$, ${\mathcal H}_t=\{A_1,\ldots ,A_t\}$, and $T=\min\{t:A_1\cup \cdots\cup A_t=V\},$ then $P({\mathcal H}_T ~\mbox{contains a perfect matching})\rightarrow 1 $ as $n\rightarrow\infty$. It is shown here that Theorem 2 follows from a conditional version of Theorem 1 that will be proved elsewhere. The key ideas in that proof are similar to those for Theorem 1, but the argument is a longer story, and it has seemed best to give the present separate proof of Theorem 1, in which those ideas may appear more clearly.

math.CO↗
arXiv2019

An isoperimetric inequality for the Hamming cube and some consequences

Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: \[ \int h_A^\beta d\mu \ge 2 \mu(A)(1-\mu(A)). \] Here $\mu$ is uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta=\log_2(3/2)$; and, for $S\subseteq V$ and $x\in V$, \[ h_S(x) = \begin{cases} d_{V \setminus S}(x) &\mbox{ if } x \in S, 0 &\mbox{ if } x \notin S \end{cases} \] (where $d_T(x)$ is the number of neighbors of $x$ in $T$). This implies inequalities involving mixtures of edge and vertex boundaries, with related stability results, and suggests some more general possibilities. One application, a stability result for the set of edges connecting two disjoint subsets of $V$ of size roughly $|V|/2$, is a key step in showing that the number of maximal independent sets in $Q_n$ is $(1+o(1))2n\exp_2[2^{n-2}]$. This asymptotic statement, whose proof will appear separately, was the original motivation for the present work.

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