arXiv subjects

Alexander Sidorenko

Publications and source records attributed to Alexander Sidorenko.

At least 19 recordsLinked to original sources

arXiv2024

Extremal numbers and Sidorenko's conjecture

Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko's conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $\mathrm{ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.

math.CO↗

arXiv2023

Turán numbers of $r$-graphs on $r+1$ vertices

Let $H_k^r$ denote an $r$-uniform hypergraph with $k$ edges and $r+1$ vertices, where $k \leq r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turán density are $π(H_k^r) \leq \frac{k-2}{r}$ for all $k \geq 3$, and $π(H_3^r) \geq 2^{1-r}$ for $k=3$. We prove that $π(H_k^r) \geq (C_k - o(1)) \, r^{-(1+\frac{1}{k-2})}$ as $r\to\infty$. In the case $k=3$, we prove $π(H_3^r) \geq (1.7215 - o(1)) \, r^{-2}$ as $r\to\infty$, and $π(H_3^r) \geq r^{-2}$ for all $r$.

math.CO↗

arXiv2023

On the asymptotic of lottery numbers

Let $L(n,k,r,p)$ denote the minimum number of $k$-subsets of an $n$-set such that all the $\binom{n}{p}$ $p$-subsets are intersected by one of them in at least $r$ elements. The case $p=r$ corresponds to the covering numbers, while the case $k=r$ corresponds to the Turán numbers. In both cases, there exists a limit of $L(n,k,r,p) / \binom{n}{r}$ as $n\to\infty$. We prove the existence of this limit in the general case.

math.CO↗

arXiv2023

Covering of triples by quadruples in bipartite and tripartite settings

Let $A$ and $B$ be disjoint sets of sizes $a$ and $b$, respectively. Let $f(a,b)$ denote the minimum number of quadruples needed to cover all triples $T \subseteq A \cup B$ such that $|T \cap A| \geq 2$. We prove upper and lower bounds on $f(a,b)$ and use them to derive upper bounds for the $(n,4,3,4)$-lottery problem.

math.CO↗

arXiv2023

Modified Erdős-Ginzburg-Ziv constants for $\mathbb{Z}_2^d$

Let $G$ be a finite abelian group written additively, and let $r$ be a multiple of its exponent. The modified Erdős-Ginzburg-Ziv constant $\mathsf{s}_r'(G)$ is the smallest integer $s$ such that every zero-sum sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We find exact values of $\mathsf{s}_{2k}'(\mathbb{Z}_2^d)$ for $d \leq 2k+1$.

math.CO↗

arXiv2023

Turán numbers $T(n,5,3)$ and graphs without induced $5$-cycles

Turán number $T(n,5,3)$ is the minimum size of a system of triples out of a base set $X$ of $n$ elements such that every quintuple in $X$ contains a triple from the system. The exact values of $T(n,5,3)$ are known for $n \leq 17$. Turán conjectured that $T(2m,5,3) = 2\binom{m}{3}$, and no counterexamples have been found so far. If this conjecture is true, then $T(2m+1,5,3) \geq \lceil m(m-2)(2m+1)/6\rceil$. We prove the matching upper bound for all $n = 2m+1 > 17$ except $n=27$.

math.CO↗

arXiv2022

On positive hypergraphs

Camarena, Csóka, Hubai, Lippner, and Lovász introduced the notion of positive graphs. This notion naturally extends to $r$-uniform hypergraphs. In the case when $r$ is odd, we prove that a hypergraph is positive if and only if its Levi graph is positive. As an application, we show that the $1$-subdivision of $K_{r,r}$ is not a positive graph when $r$ is odd.

math.CO↗

arXiv2022

On graph norms for complex-valued functions

For any given graph $H$, one may define a natural corresponding functional $\|.\|_H$ for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once $H$ is paired with a $2$-edge-colouring $α$ to assign conjugates. We say that $H$ is real-norming (resp. complex-norming) if $\|.\|_H$ (resp. $\|.\|_{H,α}$ for some $α$) is a norm on the vector space of real-valued (resp. complex-valued) functions. These generalise the Gowers octahedral norms, a widely used tool in extremal combinatorics to quantify quasirandomness. We unify these two seemingly different notions of graph norms in real- and complex-valued settings. Namely, we prove that $H$ is complex-norming if and only if it is real-norming and simply call the property norming. Our proof does not explicitly construct a suitable $2$-edge-colouring $α$ but obtains its existence and uniqueness, which may be of independent interest. As an application, we give various example graphs that are not norming. In particular, we show that hypercubes are not norming, which resolves the last outstanding problem posed in Hatami's pioneering work on graph norms.

math.CO↗

arXiv2021

Inequalities for doubly nonnegative functions

Let $g$ be a bounded symmetric measurable nonnegative function on $[0,1]^2$, and $\left\lVert g \right\rVert = \int_{[0,1]^2} g(x,y) dx dy$. For a graph $G$ with vertices $\{v_1,v_2,\ldots,v_n\}$ and edge set $E(G)$, we define \[ t(G,g) \; = \; \int_{[0,1]^n} \prod_{\{v_i,v_j\} \in E(G)} g(x_i,x_j) \: dx_1 dx_2 \cdots dx_n \; . \] We conjecture that $t(G,g) \geq \left\lVert g \right\rVert^{|E(G)|}$ holds for any graph $G$ and any function $g$ with nonnegative spectrum. We prove this conjecture for various graphs $G$, including complete graphs, unicyclic and bicyclic graphs, as well as graphs with $5$ vertices or less.

math.CO↗

arXiv2021

On Turán numbers of the complete $4$-graphs

The Turán number $T(n,α+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $α$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,α+1,r) = t_*(r) \: n^r \: α^{1-r} \: (1+o(1))$ as $α/ r \to\infty$ and $n / α\to\infty$. It is known that $t_*(2) = 1/2$, and the conjectured value of $t_*(3)$ is $2/3$. We prove that $t_*(4) < 0.706335\:$.

math.CO↗

arXiv2020

Weakly norming graphs are edge-transitive

Let $\mathcal{H}$ be the class of bounded measurable symmetric functions on $[0,1]^2$. For a function $h \in \mathcal{H}$ and a graph $G$ with vertex set $\{v_1,\ldots,v_n\}$ and edge set $E(G)$, define \[ t_G(h) \; = \; \int \cdots \int \prod_{\{v_i,v_j\} \in E(G)} h(x_i,x_j) \: dx_1 \cdots dx_n \: . \] Answering a question raised by Conlon and Lee, we prove that in order for $t_G(|h|)^{1/|E(G)|}$ to be a norm on $\mathcal{H}$, the graph $G$ must be edge-transitive.

math.CO↗

arXiv2020

On generalized Erdős-Ginzburg-Ziv constants for $\mathbb{Z}_2^d$

Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erdős-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We find exact values of $s_{2m}(\mathbb{Z}_2^d)$ for $d \leq 2m+1$. Connections to linear binary codes of maximal length and codes without a forbidden weight are discussed.

math.CO↗

arXiv2019

Approximate Steiner $(r-1,r,n)$-systems without $3$ blocks on $r+2$ points

For a family ${\mathcal F}$ of $r$-graphs, let $\mathrm{ex}(n,{\mathcal F})$ denote the maximum number of edges in an ${\mathcal F}$-free $r$-graph on $n$ vertices. Let ${\mathcal F}_r(v,e)$ denote the family of all $r$-graphs with $e$ edges and at most $v$ vertices. We prove that $\mathrm{ex}(n,{\mathcal F}_r(r+1,2) \cup {\mathcal F}_r(r+2,3)) = (\frac{1}{r} - o(1)) \binom{n}{r-1}$.

math.CO↗

arXiv2018

On Turán problems for Cartesian products of graphs

Let $A,B$ be disjoint sets of sizes $n$ and $m$. Let ${\mathcal Q}$ be a family of quadruples, having $2$ elements from $A$ and $2$ from $B$, such that any subset $S \subseteq A \cup B$ with $|S|=7$, $|S \cap A| \geq 2$ and $|S \cap B| \geq 2$ contains one of the quadruples. We prove that the smallest size of ${\mathcal Q}$ is $(1/16 + O(1/n) + O(1/m)) n^2 m^2$ as $n,m\to\infty$. We also solve asymptotically a more general two-partite Turán problem for quadruples.

math.CO↗

arXiv2018

Extremal problems on the hypercube and the codegree Turán density of complete $r$-graphs

Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erdős-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We show that $s_{2m}(\mathbb{Z}_2^d) \leq C_m 2^{d/m} + O(1)$ when $d\rightarrow\infty$, and $s_{2m}(\mathbb{Z}_2^d) \geq 2^{d/m} + 2m-1$ when $d=km$. We use results on $s_r(G)$ to prove new bounds for the codegree Turán density of complete $r$-graphs.

math.CO↗

arXiv2018

An Erdős-Gallai-type theorem for keyrings

A keyring is a graph obtained by appending $r \geq 1$ leaves to one of the vertices of a cycle. We prove that for every $r \leq (k-1)/2$, any graph with average degree more than $k-1$ contains a keyring with $r$ leaves and at least $k$ edges.

math.CO↗

arXiv2018

Spectral Estimation of Plasma Fluctuations I: Comparison of Methods

The relative root mean squared errors (RMSE) of nonparametric methods for spectral estimation is compared for microwave scattering data of plasma fluctuations. These methods reduce the variance of the periodogram estimate by averaging the spectrum over a frequency bandwidth. As the bandwidth increases, the variance decreases, but the bias error increases. The plasma spectra vary by over four orders of magnitude, and therefore, using a spectral window is necessary. We compare the smoothed tapered periodogram with the adaptive multiple taper methods and hybrid methods. We find that a hybrid method, which uses four orthogonal tapers and then applies a kernel smoother, performs best. For 300 point data segments, even an optimized smoothed tapered periodogram has a 24 \% larger relative RMSE than the hybrid method. We present two new adaptive multi-taper weightings which outperform Thomson's original adaptive weighting.

stat.AP↗

arXiv2018

Spectral Estimation of Plasma Fluctuations II: Nonstationary Analysis of ELM Spectra

Several analysis methods for nonstationary fluctuations are described and applied to the edge localized mode (ELM) instabilities of limiter H-mode plasmas. The microwave scattering diagnostic observes poloidal $k_θ$ values of 3.3 cm$^{-1}$, averaged over a 20 cm region at the plasma edge.A short autoregressive filter enhances the nonstationary component of the plasma fluctuations by removing much of the background level of stationary fluctuations. Between ELMs, the spectrum predominantly consists of broad-banded 300-700 kHz fluctuations propagating in the electron diamagnetic drift direction, indicating the presence of a negative electric field near the plasma edge. The time-frequency spectrogram is computed with the multiple taper technique. By using the singular value decomposition of the spectrogram, it is shown that the spectrum during the ELM is broader and more symmetric than that of the stationary spectrum. The ELM period and the evolution of the spectrum between ELMs varies from discharge to discharge. For the discharge under consideration which has distinct ELMs with a 1 msec period, the spectrum has a maximum in the electron drift direction which relaxes to a near constant value %its characteristic shape in the first half millisecond after the end of the ELM and then grows slowly. In contrast, the level of the fluctuations in the ion drift direction increases exponentially by a factor of eight in the five milliseconds~after the ELM. High frequency precursors are found which occur one millisecond before the ELMs and propagate in the ion drift direction. These precursors are very short ($\sim 10 μ$secs), coherent bursts, and they predict the occurrence of an ELM with a high success rate.

physics.plasm-ph↗