Questions tagged [probabilistic-method]
Probabilistic methods prove existence results in a nonconstructive fashion, by showing the chance of randomly selecting a solution is greater than zero.
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Clarification on the definition of $P(m,n)$ in Mann–Martínez (1996) "The exponent of finite groups"
In their 1996 paper "The exponent of finite groups", Mann and Martínez define a function $P(m,n)$ as:
$$P(m,n)=\text{max}\left(\frac{R(m,n^2)}{R(m,n^2)+1},1-\frac{1-P_G(n)}{R(m,n)}\right)$$
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Extremal combinatorics results matched closely by probabilistic constructions
There are numerous examples of extremal* results in combinatorics that are matched asymptotically by some probabilistic construction, but with some gap, often quite substantial. Think for example of ...
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On sets spanning all subspaces of a given dimension in vector spaces over finite fields
Let $n$ be a positive integer, let $p$ be a prime, let $\mathbb F_p$ be the field with $p$ elements, and let $V = \mathbb F_p^n$ be the $n$-dimensional vector space over $\mathbb F_p$.
For an integer $...
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Can we find background noise for every Følner sequence in a countable amenable group?
Let $G$ be a countable amenable group. We consider sequences $(z_g)_{g\in G}$ of complex numbers with $|z_g|=1$ for all $g\in G$.
I will say $(z_g)_{g\in G}$ is background noise for a (left-)Følner ...
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Quantum probabilistic method?
The probabilistic method uses arguments from probability to prove deterministic statements. This has been applied to diverse fields such as combinatorics, topology and number theory. In this method, ...
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For every partition of $E(K_{n,n})$ into $n$ colour classes of size $n$, there is a vertex incident to at least $\sqrt{n}$ colours
Given a partition of the edges of $K_{n,n}$ into $n$ colours, where each colour appears exactly $n$ times, prove that there exists a vertex incident to at least $\sqrt{n}$ colours.
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Probabilistic method Alon and Spencer Azuma's inequality
Theorem 7.5.2 states:
Let $v_1, \dots, v_n$ be vectors with $\|v_i\| \leq 1.$ Let $\epsilon_1, \dots, \epsilon_n \in \{-1, 1\}$ be independent with uniform probability and let $X=\|\epsilon_1 v_1 + \...
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Existence of (near) equidistant codewords
My question is originally related to coding theory, but fairly easy to state in pure combinatorial way.
Fix $k\in\mathbb{N}$, $\beta\in(0,1)$ and consider the binary cube $\Sigma_n = \{0,1\}^n$ ...
user127150
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Why subcopula is less used in modelling?
I don't know if it is a good idea to post my question in MathOverflow instead of Mathexchange. But it seems to me that it is more appropriate to post my question in MathOverflow.
By definition, copula ...
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Counting progressions for Ramsey-type number
I'm currently learning about Ramsey theory and how to use the probabilistic method to find lower bounds. Currently I'm looking at a family, which I'll call $H$, that is composed of the following ...
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Probabilistic method with multiple objective functions
Let $\mathcal X$ be a finite set and $f$ a function from $\mathcal X$ to $\mathbb R^+$. Basic probabilistic method says that if I can find a probability distribution on $\mathcal X$ and show that $E[f(...
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Theorems like the Lovász Local Lemma?
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have ...
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On an exercise in The Probabilistic Method : random dilate of a set in a finite field
This is related to Problem $4.6$ in ``The Probabilistic Method'' by Alon and Spencer, where one essentially has to prove the following:
Let $p$ be a prime, and $A$ be any subset of $\mathbb{F}_p$. ...
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Domination problem with sets
For nearly two years, I have been struggling with the next task I have already published on MSE, but unfortunately with no respond.
Let $M$ be a non-empty and finite set, $S_1,...,S_k$ subsets
of ...
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$\lim_{k\to\infty}{n\choose k}2^{1-{k\choose2}}$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$
How do you prove $\lim_{k\to\infty,k\in\mathbb{N}}{n\choose k}2^{1-{k\choose2}}=1$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$? The expression ${n\choose k}2^{1-{k\choose2}}$ ...
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A Non-trivial intersecting set system problem
Liven large enough $k\in\Bbb N$ fix $m\in\{2,3,\dots,k\}$ and fix $4k$ cardinality set $K_{4k}$.
What is the maximum $n\in\Bbb N$ such that at some $t\geq2n-1$ there are $$\mbox{ subsets }L_1,L_2,\...
user94040
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Set theoretic forcing, large cardinals and probabilistic methods
This question is motivated by the work of Ajtai "The complexity of the pigeonhole principle" and similar works. In this paper, the author proves that $PHP_n$, the pigeonhole principle for $...
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An extremal combinatorics problem involving column summation
Given $n\in\Bbb N$, $\alpha>0$, $\beta\in\big[\frac12,1\big]$ denote $\mathcal R_{n,\alpha,\beta}$ as collection of all $2^n\times 2^{n^\alpha}$ $0/1$ matrices with every row summing to strictly ...
user76479
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What are fun elementary subjects in probability?
I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....
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Does $|A+A|$ concentrate near its mean?
Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...
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Derandomization barriers in complexity theory applicable as barriers to constructive arguments replacing probabilistic method
The probabilistic method as first pioneered by Erdős (although others used this before) shows existence of a certain object while finding that object may take exponential time.
(1) Is there any ...
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List of proofs where existence through probabilistic method has not been constructivised
The probabilistic method as first pioneered by Erdős (although others have used this before) shows the existence of a certain object. What are some of the most important objects for which we can show ...
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Assigning random orientation to an edge in a regular graph
Given a simple regular graph of degree $d$ on $n$ vertices.
Assume an ordering of vertices and assume all orientations of edges is from $i$ to $j$ if edges $ij$ exists and $i<j$. Pick $m$ random ...
user76479
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Maximal number of subsets in $\{1,\dots,n\}$ such that neither is contained in a union of two others
What are known estimates for maximal $M$ for which their exists subsets $A_1,\dots,A_M$ in $\{1,\dots,n\}$ such that there do not exist different indexes $i,j,k$ for which $A_i\subset A_j\cup A_k$?
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