arXiv subjects

David Ellis

Publications and source records attributed to David Ellis.

53 records · Page 3Linked to original sources

arXiv2017

A quasi-stability result for dictatorships in $S_{n}$

We prove that Boolean functions on $S_{n}$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_n$, are close to being unions of cosets of point-stabilizers. We use this to give a natural proof of a stability result on intersecting families of permutations, originally conjectured by Cameron and Ku, and first proved by the first author. We also use it to prove a `quasi-stability' result for an edge-isoperimetric inequality in the transposition graph on $S_n$, namely that subsets of $S_n$ with small edge-boundary in the transposition graph are close to being unions of cosets of point-stabilizers.

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arXiv2017

A stability result for balanced dictatorships in $S_{n}$

We prove that a balanced Boolean function on $S_{n}$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_{n}$, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on $S_{n}$ generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from $0$ and $1$. In contrast, [Ellis, D., Filmus, Y., Friedgut, E., A quasi-stability result for dictatorships in $S_{n}$, Combinatorica 35 (2015), pp. 573-618] deals with Boolean functions of expectation O(1/n) whose Fourier transform is highly concentrated on the first two irreducible representations of $S_{n}$. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.

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arXiv2017

Low-degree Boolean functions on $S_n$, with an application to isoperimetry

We prove that Boolean functions on $S_n$, whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of $n$ whose largest part has size at least $n-t$, are close to being unions of cosets of stabilizers of $t$-tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on $S_n$ which is asymptotically sharp for subsets of $S_n$ of size $n!/\textrm{poly}(n)$, using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of $S_n$ of size $(n-t)!$, where $n$ is large compared to $t$, confirming a conjecture of Ben Efraim in these cases.

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arXiv2017

On symmetric 3-wise intersecting families

A family of sets is said to be symmetric if its automorphism group is transitive, and $3$-wise intersecting if any three sets in the family have nonempty intersection. Frankl conjectured in 1981 that if $\mathcal{A}$ is a symmetric $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$, then $|\mathcal{A}| = o(2^n)$. Here, we give a short proof of Frankl's conjecture using a 'sharp threshold' result of Friedgut and Kalai.

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arXiv2017

Geometric stability via information theory

The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. In this paper, we prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close in symmetric difference to a 'box'. Our results are best possible up to a constant factor depending upon the dimension alone. Our approach is information theoretic. We use our stability result for the Loomis-Whitney inequality to obtain a stability result for the edge-isoperimetric inequality in the infinite $d$-dimensional lattice. Namely, we prove that a subset of $\mathbb{Z}^d$ with small edge-boundary must be close in symmetric difference to a $d$-dimensional cube. Our bound is, again, best possible up to a constant factor depending upon $d$ alone.

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arXiv2016

On the structure of graphs which are locally indistinguishable from a lattice

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional square lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d \geq 3$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They may be viewed as quotient lattices of $\mathbb{L}^d$ in various compact orbifolds.) In the $d=2$ case, our methods yield new proofs of structure theorems of Thomassen and of Márquez, de Mier, Noy and Revuelta, and also yield short, `algebraic' restatements of these theorems. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory.

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arXiv2015

Juntas in the $\ell^{1}$-grid and Lipschitz maps between discrete tori

We show that if $A \subset [k]^n$, then $A$ is $ε$-close to a junta depending upon at most $\exp(O(|\partial A|/(k^{n-1}ε)))$ coordinates, where $\partial A$ denotes the edge-boundary of $A$ in the $\ell^1$-grid. This is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [E. Friedgut, Boolean functions with low average sensitivity depend on few coordinates, Combinatorica 18 (1998), 27-35], or as a characterization of large subsets of the $\ell^1$-grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [T. Austin, On the failure of concentration for the $\ell^{\infty}$-ball, preprint]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.

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arXiv2014

An isoperimetric inequality for conjugation-invariant sets in the symmetric group

We prove an isoperimetric inequality for conjugation-invariant sets of size $k$ in $S_n$, showing that these necessarily have edge-boundary considerably larger than some other sets of size $k$ (provided $k$ is small). Specifically, let $T_n$ denote the Cayley graph on $S_n$ generated by the set of all transpositions. We show that if $A \subset S_n$ is a conjugation-invariant set with $|A| = pn! \leq n!/2$, then the edge-boundary of $A$ in $T_n$ has size at least $$c \cdot \frac {\log_2 (\tfrac 1{p})}{\log_2 \log_2 (\tfrac 2{p})}\cdot n \cdot |A|,$$ where $c$ is an absolute constant. (This is sharp up to an absolute constant factor, when $p = Θ(1/s!)$ for any $s \in \{1,2,...,n\}$.) It follows that if $p = n^{-Θ(1)}$, then the edge-boundary of a conjugation-invariant set of measure $p$ is necessarily a factor of $Ω(\log n / \log \log n)$ larger than the minimum edge-boundary over all sets of measure $p$.

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arXiv2013

Almost isoperimetric subsets of the discrete cube

We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + ε)$ can be made into a subcube by at most $(2 ε/\log_{2}(1/ε))|A|$ additions and deletions, provided $ε$ is less than an absolute constant. We deduce that if $A \subset \{0,1\}^{n}$ has size $2^{t}$ for some $t \in \mathbb{N}$, and cannot be made into a subcube by fewer than $δ|A|$ additions and deletions, then its edge-boundary has size at least $|A| \log_{2}(2^{n}/|A|) + |A| δ\log_{2}(1/δ) = 2^{t}(n-t+δ\log_{2}(1/δ))$, provided $δ$ is less than an absolute constant. This is sharp whenever $δ= 1/2^{j}$ for some $j \in \{1,2,\ldots,t\}$.

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arXiv2013

Forbidding just one intersection, for permutations

We prove that for $n$ sufficiently large, if $A$ is a family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly once, then $|\mathcal{A}| \leq (n-2)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of 2 points. We also obtain a Hilton-Milner type result, namely that if $\mathcal{A}$ is such a family which is not contained within a coset of the stabilizer of 2 points, then it is no larger than the family $\{σ\in S_{n}:\ σ(1)=1,σ(2)=2,\ \#\{\textrm{fixed points of}σ\geq 5\} \neq 1\} \cup \{(1\ 3)(2\ 4),(1\ 4)(2\ 3),(1\ 3\ 2\ 4),(1\ 4\ 2\ 3)\}$. We conjecture that for $t \in \mathbb{N}$, and for $n$ sufficiently large depending on $t$, if $\mathcal{A}$ is family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly $t-1$ times, then $|\mathcal{A}| \leq (n-t)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of $t$ points. This can be seen as a permutation analogue of a conjecture of Erdős on families of $k$-element sets with a forbidden intersection, proved by Frankl and Füredi in [P. Frankl and Z. Füredi, Forbidding Just One Intersection, Journal of Combinatorial Theory, Series A, Volume 39 (1985), pp. 160-176].

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arXiv2012

Triangle-Intersecting Families of Graphs

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.

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arXiv2012

An approximate isoperimetric inequality for r-sets

We prove a vertex-isoperimetric inequality for [n]^(r), the set of all r-element subsets of {1,2,...,n}, where x,y \in [n]^(r) are adjacent if |x Δy|=2. Namely, if \mathcal{A} \subset [n]^(r) with |\mathcal{A}|=α{n \choose r}, then the vertex-boundary b(\mathcal{A}) satisfies |b(\mathcal{A})| \geq c\sqrt{\frac{n}{r(n-r)}} α(1-α) {n \choose r}, where c is a positive absolute constant. For αbounded away from 0 and 1, this is sharp up to a constant factor (independent of n and r).

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arXiv2011

Generating all subsets of a finite set with disjoint unions

If X is an n-element set, we call a family G of subsets of X a k-generator for X if every subset of X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo conjectured that for n > 2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjecture for all sufficiently large n when k = 2, and for n a sufficiently large multiple of k when k > 2.

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arXiv2011

A Proof of the Cameron-Ku conjecture

A family of permutations A \subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any σ, π\in A, there is some i such that σ(i)=π(i). Deza and Frankl showed that for such a family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A = {σ\in S_{n}: σ(i)=j} for some i and j. They conjectured a `stability' version of this result, namely that there exists a constant c < 1 such that if A \subset S_{n} is an intersecting family of size at least c(n-1)!, then there exist i and j such that every permutation in A maps i to j (we call such a family `centred'). They also made the stronger `Hilton-Milner' type conjecture that for n \geq 6, if A \subset S_{n} is a non-centred intersecting family, then A cannot be larger than the family C = {σ\in S_{n}: σ(1)=1, σ(i)=i \textrm{for some} i > 2} \cup {(12)}, which has size (1-1/e+o(1))(n-1)!. We prove the stability conjecture, and also the Hilton-Milner type conjecture for n sufficiently large. Our proof makes use of the classical representation theory of S_{n}. One of our key tools will be an extremal result on cross-intersecting families of permutations, namely that for n \geq 4, if A,B \subset S_{n} are cross-intersecting, then |A||B| \leq ((n-1)!)^{2}. This was a conjecture of Leader; it was recently proved for n sufficiently large by Friedgut, Pilpel and the author.

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arXiv2010

Irredundant Families of Subcubes

We consider the problem of finding the maximum possible size of a family of k-dimensional subcubes of the n-cube {0,1}^{n}, none of which is contained in the union of the others. (We call such a family `irredundant'). Aharoni and Holzman conjectured that for k > n/2, the answer is {n choose k} (which is attained by the family of all k-subcubes containing a fixed point). We give a new proof of a general upper bound of Meshulam, and we prove that for k >= n/2, any irredundant family in which all the subcubes go through either (0,0,...,0) or (1,1,...,1) has size at most {n choose k}. We then give a general lower bound, showing that Meshulam's upper bound is always tight up to a factor of at most e.

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arXiv2009

Stability for t-intersecting families of permutations

A family of permutations (\mathcal{A} \subset S_{n}) is said to be (t)-\textit{intersecting} if any two permutations in (\mathcal{A}) agree on at least (t) points, i.e. for any (σ, π\in \mathcal{A}), (|\{i \in [n]: σ(i)=π(i)\}| \geq t). It was recently proved by Friedgut, Pilpel and the author that for (n) sufficiently large depending on (t), a (t)-intersecting family (\mathcal{A} \subset S_{n}) has size at most ((n-t)!), with equality only if (\mathcal{A}) is a coset of the stabilizer of (t) points (or `(t)-coset' for short), proving a conjecture of Deza and Frankl. Here, we first obtain a rough stability result for (t)-intersecting families of permutations, namely that for any (t \in \mathbb{N}) and any positive constant (c), if (\mathcal{A} \subset S_{n}) is a (t)-intersecting family of permutations of size at least (c(n-t)!), then there exists a (t)-coset containing all but at most a (O(1/n))-fraction of (\mathcal{A}). We use this to prove an exact stability result: for (n) sufficiently large depending on (t), if (\mathcal{A} \subset S_{n}) is a (t)-intersecting family which is not contained within a (t)-coset, then (\mathcal{A}) is at most as large as the family \mathcal{D} & = & \{σ\in S_{n}: σ(i)=i \forall i \leq t, σ(j)=j \textrm{for some} j > t+1\} && \cup \{(1 t+1),(2 t+1),...,(t t+1)\} which has size ((1-1/e+o(1))(n-t)!). Moreover, if (\mathcal{A}) is the same size as (\mathcal{D}) then it must be a `double translate' of (\mathcal{D}), meaning that there exist (π,τ\in S_{n}) such that (\mathcal{A}=π\mathcal{D} τ). We also obtain an analogous result for (t)-intersecting families in the alternating group (A_{n}).

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arXiv2008

Note on generating all subsets of a finite set with disjoint unions

We call a family G of subsets of [n] a k-generator of (\mathbb{P}[n]) if every (x \subset [n]) can be expressed as a union of at most k disjoint sets in (\mathcal{G}). Frein, Leveque and Sebo conjectured that for any (n \geq k), such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize a theorem of Alon and Frankl \cite{alon} in order to show that for fixed k, any k-generator of (\mathbb{P}[n]) must have size at least (k2^{n/k}(1-o(1))), thereby verifying the conjecture asymptotically for multiples of k.

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