arXiv subjects

Elad Aigner-Horev

Publications and source records attributed to Elad Aigner-Horev.

At least 19 recordsLinked to original sources

arXiv2026

Rainbow spanning configurations in uniformly coloured pseudorandom graphs

We prove a quantitative palette-transference principle for rainbow spanning configurations in uniformly edge-coloured pseudorandom graphs. The input to our transference principle is an embedding result of a spanning configuration in an appropriately bijumbled graph $H$ with sufficiently large minimum degree. The output of our transference principle is the asymptotically almost sure existence of a rainbow copy of the same configuration in a uniformly edge-coloured graph $G$ whose bijumbledness and minimum are comparable and sometimes coincide with those of $H$. We then apply our transference principle in order to asymptotically almost surely obtain $K_k$-factors, including perfect matchings, Hamilton cycles, and a prescribed bounded-degree spanning tree in bijumbled graphs with appropriate parameters. In all of our results, the palette size exceeds the size of the target configuration by $\varepsilon n$, where $\varepsilon > 0$ is arbitrarily small yet fixed, and $n$ is the order of the configuration.

math.CO↗

arXiv2026

Resilience of rainbow Hamilton cycles in pseudorandom graphs

For every fixed $\varepsilon\in(0,1/2)$, we prove that every spanning subgraph $H$ of an $n$-vertex $(p,β)$-bijumbled graph satisfying $δ(H)\geq(1/2+\varepsilon)pn$ contains a rainbow Hamilton cycle under every globally $μpn$-bounded colouring, provided $β\leq cpn$ and $pn\geq M$ both hold. The same assertion holds under the relative condition $°_H(v)\geq(1/2+\varepsilon)°_G(v)$ for every vertex $v$, provided $δ(G)\geq(1-\varepsilon/4)pn$ holds. Under either degree condition, there are at least $(apn)^n$ such cycles. Here, $c,μ,a,M>0$ depend only on $\varepsilon$; in particular, $pn$ may be a sufficiently large constant. If $pn\geq D\log n$, then, upon fixing a coloured $H$, retaining each edge independently with probability $D\log n/(pn)$ preserves rainbow Hamiltonicity asymptotically almost surely. The logarithmic degree requirement is needed only for this percolation conclusion. The existence theorem answers a problem of Coulson, Keevash, Perarnau and Yepremyan for random graphs and extends it to deterministic pseudorandom hosts. In fact, all three conclusions hold with rainbowness replaced by avoidance of prescribed pairs of edges; each edge having at most $μpn$ conflicting partners. We construct an $O(1/(pn))$-spread probability measure on conflict-free Hamilton cycles; this then yields the enumeration and percolation results.

math.CO↗

arXiv2026

The Sample Complexity of Distributionally Robust PAC Learning under Cressie--Read Divergences

We study distributionally robust PAC learning for the $0$--$1$-loss, where adversarial perturbations of the data distribution are constrained by a Cressie--Read divergence of order $k>1$ and radius $ρ\geq 0$. For hypothesis classes with VC dimension $d$, we establish realizable and agnostic sample-complexity bounds tight up to constant and logarithmic factors, respectively; ordinary empirical risk minimization attains both rates up to logarithmic factors. For target accuracy $\varepsilon\in(0,1)$ and confidence $δ\in(0,1)$, their respective orders are \[ \max\!\left\{\frac{1}{\varepsilon}, \frac{ρ^{\frac 1{k-1}}}{\varepsilon^{k_\star}} \right\}\cdot(d+\log δ^{-1}) \qquad\text{and}\qquad \max\!\left\{\frac{1}{\varepsilon^2}, \frac{ρ^{\frac1{k-1}}}{\varepsilon^{k_\star\vee 2}} \right\}\cdot(d+\log δ^{-1}), \] where $k_\star={k}/{(k-1)}$. For every fixed $ρ>0$, robustness changes the realizable $\varepsilon$-dependence from $\varepsilon^{-1}$ to $\varepsilon^{-k_\star}$ as $\varepsilon\downarrow0$. In the agnostic case, for $1 1$, close its upper--lower gaps, and recover standard PAC learning rates as $ρ\to0$, unlike previous bounds that fail to interpolate correctly in this limit.

cs.LG↗

arXiv2025

Resilience of Rademacher chaos of low degree

The {\em resilience} of a Rademacher chaos is the maximum number of adversarial sign-flips that the chaos can sustain without having its largest atom probability significantly altered. Inspired by probabilistic lower-bound guarantees for the resilience of linear Rademacher chaos (aka. resilience of the Littlewood-Offord problem), obtained by Bandeira, Ferber, and Kwan (Advances in Mathematics, Vol. $319$, $2017$), we provide probabilistic lower-bound guarantees for the resilience of Rademacher chaos of arbitrary degree; these being most meaningful provided that the degree is constant.

math.PR↗

arXiv2025

Smoothed analysis in compressed sensing

Arbitrary matrices $M \in \mathbb{R}^{m \times n}$, randomly perturbed in an additive manner using a random matrix $R \in \mathbb{R}^{m \times n}$, are shown to asymptotically almost surely satisfy the so-called {\sl robust null space property}. Whilst insisting on an asymptotically optimal order of magnitude for $m$ required to attain {\sl unique reconstruction} via $\ell_1$-minimisation algorithms, our results track the level of arbitrariness allowed for the fixed seed matrix $M$ as well as the degree of distributional irregularity allowed for the entries of the perturbing matrix $R$. Starting with sub-gaussian entries for $R$, our results culminate with these allowed to have substantially heavier tails than sub-exponential ones. Throughout this trajectory, two measures control the arbitrariness allowed for $M$; the first is $\|M\|_\infty$ and the second is a localised notion of the Frobenius norm of $M$ (which depends on the sparsity of the signal being reconstructed). A key tool driving our proofs is {\sl Mendelson's small-ball method} ({\em Learning without concentration}, J. ACM, Vol. $62$, $2015$).

math.PR↗

arXiv2025

Ramsey properties of randomly perturbed hypergraphs

We study Ramsey properties of randomly perturbed $3$-uniform hypergraphs. For~$t\geq 2$, write $\tilde K^{(3)}_t$ to denote the $3$-uniform {\it expanded} clique hypergraph obtained from the complete graph $K_t$ by expanding each of the edges of the latter with a new additional vertex. For an even integer $t\geq 4$, let~$M$ denote the asymmetric maximal density of the pair $(\tilde K^{(3)}_t,\tilde K^{(3)}_{t/2})$. We prove that adding a set~$F$ of random hyperedges satisfying $|F|\gg n^{3-1/M}$ to a given $n$-vertex $3$-uniform hypergraph~$H$ with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for $\tilde K^{(3)}_t$ and two colours. We conjecture that this result is asymptotically best possible with respect to the size of $F$ whenever $t\geq 6$ is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a \emph{tuple lemma} providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.

math.CO↗

arXiv2024

Smoothed Analysis of the Komlós Conjecture: Rademacher Noise

The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Komlós, stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Komlós matrix, that is, whenever every column of $M$ lies within the unit sphere. Our main result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ holds asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is Komlós, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d = ω(1)$, and $n = ω(d \log d)$. The factor $d^{-1/2}$ normalising $R$ is essentially best possible and the dependency between $n$ and $d$ is asymptotically best possible. Our main source of inspiration is a result by Bansal, Jiang, Meka, Singla, and Sinha (ICALP 2022). They obtained an assertion similar to the one above in the case that the smoothing matrix is Gaussian. They asked whether their result can be attained with the optimal dependency $n = ω(d \log d)$ in the case of Bernoulli random noise or any other types of discretely distributed noise; the latter types being more conducive for Smoothed Analysis in other discrepancy settings such as the Beck-Fiala problem. For Bernoulli noise, their method works if $n = ω(d^2)$. In the case of Rademacher noise, we answer the question posed by Bansal, Jiang, Meka, Singla, and Sinha. Our proof builds upon their approach in a strong way and provides a discrete version of the latter. Breaking the $n = ω(d^2)$ barrier and reaching the optimal dependency $n = ω(d \log d)$ for Rademacher noise requires additional ideas expressed through a rather meticulous counting argument, incurred by the need to maintain a high level of precision all throughout the discretisation process.

math.CO↗

arXiv2022

Minors, connectivity, and diameter in randomly perturbed sparse graphs

Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every $n$-vertex graph $G$ contains a complete minor of order $Ω(n/α(G))$. We prove that adding $ξn$ random edges, where $ξ> 0$ is arbitrarily small yet fixed, to an $n$-vertex graph $G$ satisfying $α(G) \leq ζ(ξ)n$ asymptotically almost surely results in a graph containing a complete minor of order $\tilde Ω\left( n/\sqrt{α(G)}\right)$; this result is tight up to the implicit logarithmic terms. For complete topological minors, we prove that there exists a constant $C>0$ such that adding $C n$ random edges to a graph $G$ satisfying $δ(G) = ω(1)$, asymptotically almost surely results in a graph containing a complete topological minor of order $\tilde Ω(\min\{δ(G),\sqrt{n}\})$; this result is tight up to the implicit logarithmic terms. Finally, extending results of Bohman, Frieze, Krivelevich, and Martin for the dense case, we analyse the asymptotic behaviour of the vertex-connectivity and the diameter of randomly perturbed sparse graphs.

math.CO↗

arXiv2022

Large rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every {\sl proper} colouring of its edges yields a {\sl rainbow} copy of $H$. We study the thresholds for such so-called {\sl anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d$, and $d$ is a constant that does not depend on $n$. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_s$ for every $s$. In this paper, we show that for $s \geq 9$ the threshold is $n^{-1/m_2(K_{\left\lceil s/2 \right\rceil})}$; in fact, our $1$-statement is a supersaturation result. This turns out to (almost) be the threshold for $s=8$ as well, but for every $4 \leq s \leq 7$, the threshold is lower; see our companion paper for more details. In this paper, we also consider the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} C_{2\ell - 1}$, and show that the threshold for this property is $n^{-2}$ for every $\ell \geq 2$; in particular, it does not depend on the length of the cycle $C_{2\ell - 1}$. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.

math.CO↗

arXiv2022

Small rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every \emph{proper} colouring of its edges yields a \emph{rainbow} copy of $H$. We study the thresholds for such so-called \emph{anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d >0$, and $d$ is independent of $n$. In a companion article, we proved that the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_\ell$ is $n^{-1/m_2(K_{\left\lceil \ell/2 \right\rceil})}$, whenever $\ell \geq 9$. For smaller $\ell$, the thresholds behave more erratically, and for $4 \le \ell \le 7$ they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for \emph{large} cliques. In particular, we show that the thresholds for $\ell \in \{4, 5, 7\}$ are $n^{-5/4}$, $n^{-1}$, and $n^{-7/15}$, respectively. For $\ell \in \{6, 8\}$ we determine the threshold up to a $(1 + o(1))$-factor in the exponent: they are $n^{-(2/3 + o(1))}$ and $n^{-(2/5 + o(1))}$, respectively. For $\ell = 3$, the threshold is $n^{-2}$; this follows from a more general result about odd cycles in our companion paper.

math.CO↗

arXiv2022

Cycle lengths in randomly perturbed graphs

Let $G$ be an $n$-vertex graph, where $δ(G) \geq δn$ for some $δ:= δ(n)$. A result of Bohman, Frieze and Martin from 2003 asserts that if $α(G) = O \left(δ^2 n \right)$, then perturbing $G$ via the addition of $ω\left(\frac{\log(1/δ)}{δ^3} \right)$ random edges, asymptotically almost surely (a.a.s. hereafter) results in a Hamiltonian graph. This bound on the size of the random perturbation is only tight when $δ$ is independent of $n$ and deteriorates as to become uninformative when $δ= Ω\left(n^{-1/3} \right)$. We prove several improvements and extensions of the aforementioned result. First, keeping the bound on $α(G)$ as above and allowing for $δ= Ω(n^{-1/3})$, we determine the correct order of magnitude of the number of random edges whose addition to $G$ a.a.s. results in a pancyclic graph. Our second result ventures into significantly sparser graphs $G$; it delivers an almost tight bound on the size of the random perturbation required to ensure pancyclicity a.a.s., assuming $δ(G) = Ω\left((α(G) \log n)^2 \right)$ and $α(G) δ(G) = O(n)$. Assuming the correctness of Chvátal's toughness conjecture, allows for the mitigation of the condition $α(G) = O \left(δ^2 n \right)$ imposed above, by requiring $α(G) = O(δ(G))$ instead; our third result determines, for a wide range of values of $δ(G)$, the correct order of magnitude of the size of the random perturbation required to ensure the a.a.s. pancyclicity of $G$. For the emergence of nearly spanning cycles, our fourth result determines, under milder conditions, the correct order of magnitude of the size of the random perturbation required to ensure that a.a.s. $G$ contains such a cycle.

math.CO↗

arXiv2022

Envy-free Matchings in Bipartite Graphs and their Applications to Fair Division

A matching in a bipartite graph with parts X and Y is called envy-free if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources ("cakes") or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-floor(2n/3) maximin-share allocation of discrete bads among n agents.

cs.DS↗

arXiv2021

Rainbow trees in uniformly edge-coloured graphs

We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform colouring of $\mathbb{G}(n,ω(1)/n)$, using a palette of size $n$, a.a.s. admits a rainbow copy of any given bounded-degree tree on at most $(1-\varepsilon)n$ vertices, where $\varepsilon > 0$ is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaining to the embedding of bounded-degree almost-spanning prescribed trees in $\mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$. Given an $n$-vertex graph $G$ with minimum degree at least $δn$, where $δ> 0$ is fixed, we use our aforementioned result in order to prove that a uniform colouring of the randomly perturbed graph $G \cup \mathbb{G}(n,ω(1)/n)$, using $(1+α)n$ colours, where $α> 0$ is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree {\sl spanning} tree. This can be viewed as a rainbow variant of a result by Krivelevich, Kwan, and Sudakov who proved that $G \cup \mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$, a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with $G$ as above, we prove that a uniform colouring of $G \cup \mathbb{G}(n,ω(n^{-2}))$ using $n-1$ colours a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

math.CO↗

arXiv2020

On sparse random combinatorial matrices

Let $Q_{n,d}$ denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at random from the subset of vectors in $\{0,1\}^n$ having precisely $d$ entries equal to $1$. We present a short proof of the fact that $\Pr[\det(Q_{n,d})=0] = O\left(\frac{n^{1/2}\log^{3/2} n}{d}\right)=o(1)$, whenever $d=ω(n^{1/2}\log^{3/2} n)$. In particular, our proof accommodates sparse random combinatorial matrices in the sense that $d = o(n)$ is allowed. We also consider the singularity of deterministic integer matrices $A$ randomly perturbed by a sparse combinatorial matrix. In particular, we prove that $\Pr[\det(A+Q_{n,d})=0]=O\left(\frac{n^{1/2}\log^{3/2} n}{d}\right)$, again, whenever $d=ω(n^{1/2}\log^{3/2} n)$ and $A$ has the property that $(1,-d)$ is not an eigenpair of $A$.

math.CO↗

arXiv2020

Rainbow Hamilton cycles in randomly coloured randomly perturbed dense graphs

Given an $n$-vertex graph $G$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $G \cup \mathbb{G}(n,p)$ over the supergraphs of $G$ is referred to as a (random) {\sl perturbation} of $G$. We consider the distribution of edge-coloured graphs arising from assigning each edge of the random perturbation $G \cup \mathbb{G}(n,p)$ a colour, chosen independently and uniformly at random from a set of colours of size $r := r(n)$. We prove that such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) \geq C/n$, for some fixed $C > 0$, and $r = (1 + o(1))n$. The number of colours used is clearly asymptotically best possible. In particular, this improves upon a recent result of Anastos and Frieze (2019) in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-coloured sparse pseudo-random graphs a.a.s. admit an almost spanning rainbow path.

math.CO↗

arXiv2019

An Asymmetric Random Rado Theorem: 1-statement

A classical result by Rado characterises the so-called partition-regular matrices $A$, i.e.\ those matrices $A$ for which any finite colouring of the positive integers yields a monochromatic solution to the equation $Ax=0$. We study the {\sl asymmetric} random Rado problem for the (binomial) random set $[n]_p$ in which one seeks to determine the threshold for the property that any $r$-colouring, $r \geq 2$, of the random set has a colour $i \in [r]$ admitting a solution for the matrical equation $A_i x = 0$, where $A_1,\ldots,A_r$ are predetermined partition-regular matrices pre-assigned to the colours involved. We prove a $1$-statement for the asymmetric random Rado property. In the symmetric setting our result retrieves the $1$-statement of the {\sl symmetric} random Rado theorem established in a combination of results by Rödl and Ruciński~\cite{RR97} and by Friedgut, Rödl and Schacht~\cite{FRS10}. We conjecture that our $1$-statement in fact unveils the threshold for the asymmetric random Rado property, yielding a counterpart to the so-called {\em Kohayakawa-Kreuter conjecture} concerning the threshold for the asymmetric random Ramsey problem in graphs. We deduce the aforementioned $1$-statement for the asymmetric random Rado property after establishing a broader result generalising the main theorem of Friedgut, Rödl and Schacht from~\cite{FRS10}. The latter then serves as a combinatorial framework through which $1$-statements for Ramsey-type problems in random sets and (hyper)graphs alike can be established in the asymmetric setting following a relatively short combinatorial examination of certain hypergraphs. To establish this framework we utilise a recent approach put forth by Mousset, Nenadov and Samotij~\cite{MNS18} for the Kohayakawa-Kreuter conjecture.

math.CO↗

arXiv2018

Monochromatic Schur triples in randomly perturbed dense sets of integers

Given a dense subset $A$ of the first $n$ positive integers, we provide a short proof showing that for $p=ω(n^{-2/3})$ the so-called {\sl randomly perturbed} set $A \cup [n]_p$ a.a.s. has the property that any $2$-colouring of it has a monochromatic Schur triple, i.e.\ a triple of the form $(a,b,a+b)$. This result is optimal since there are dense sets $A$, for which $A\cup [n]_p$ does not possess this property for $p=o(n^{-2/3})$.

math.CO↗

arXiv2016

Extremal results for odd cycles in sparse pseudorandom graphs

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $Γ$ the generalized Turán density $π_F(Γ)$ denotes the density of a maximum subgraph of $Γ$, which contains no copy of~$F$. Extending classical Turán type results for odd cycles, we show that $π_{F}(Γ)=1/2$ provided $F$ is an odd cycle and $Γ$ is a sufficiently pseudorandom graph. In particular, for $(n,d,λ)$-graphs $Γ$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-λ,λ]$, our result holds for odd cycles of length $\ell$, provided \[ λ^{\ell-2}\ll \frac{d^{\ell-1}}n\log(n)^{-(\ell-2)(\ell-3)}\,. \] Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabó, and Vu, who addressed the case when $F$ is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free $(n,d,λ)$-graphs) shows that our assumption on $Γ$ is best possible up to the polylog-factor for every odd $\ell\geq 5$.

math.CO↗