All Questions
Tagged with extremal-graph-theory ramsey-theory
14 questions
1
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1
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159
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Acyclic partition of edges in tournaments
The following question is related to a research problem I am working on. I am curious if anyone is aware of a solution, if there are similar problems which may aid me in finding a solution, or if the ...
- 23.7k
4
votes
0
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113
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What properties do graphs avoiding large regular subgraphs have?
Fix a positive integer $r$ and real $\delta \in (0,1)$.
Let $G$ be an undirected graph on $n$ vertices. Suppose that $G$ does not contain an $r$-regular subgraph on at least $\delta n$ vertices (i.e., ...
- 557
6
votes
2
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317
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Is there a graph that is Ramsey for $P_{2n}$ but is $C_{2n+1}-$free
Write $F\to G$ to mean that for every two coloring of the edges of $F$, there exists a monochromatic copy of $G$. Nešetřil and Rödl proved that for every graph $G$, there exists a graph $F$ such that ...
- 32.1k
15
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4
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639
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Sets of points containing permutations - a Ramsey-type question
The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with ...
- 32.1k
7
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2
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362
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Reference request: monochromatic paths in edge-colored complete graphs
Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$.
...
- 32.1k
11
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1
answer
396
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Dense triangle-free graphs and their independent sets
Recall that a graph is triangle-free if it does not contain a copy of $K_3$. Also, for a graph $G$, $\alpha(G)$ shall denote its independence number. Lastly, we will write $o(1)$ to denote quantities ...
- 306
3
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2
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276
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Ramsey-Turán density function is well defined
Define
$$RT(n,K_l,f(n))=ex_l(n,f(n))=\max_G\{e(G): K_l \not\subset G, v(G)=n, \alpha(G)\leq f(n)\}$$
and the Ramsey-Turán density function $f_l:(0,1] \to \mathbb{R}$ as
$$f_l(\alpha)=\lim_{n\to \infty}...
- 211
4
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2
answers
2k
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The number of monochromatic triangles
It is well known that the minimum number of monochromatic triangles in a red/blue coloring of the edges of the complete graph $K_n$ is given by Goodman's formula
$$M(n)=\binom n3-\left\lfloor\frac n2\...
- 7,013
4
votes
1
answer
230
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Independence number of $C_4$-free graphs
It's well known that a $C_4$-free graph of order $n$ has average degree $O(\sqrt{n})$, and it follows that the independence number is $\Omega(\sqrt{n})$.
This bound cannot be improved over $\Theta(n^{\...
- 109k
6
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108
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What is the smallest number of vertices in a graph whose every orientation contains a directed straight path of length 3
For a graph $\Gamma$ and a digraph $\vec H$ we write $\Gamma\Rightarrow \vec H$ if any orientation of $\Gamma$ contains an isometric and isomorphic copy of the digraph $\vec H$. Since each graph ...
CommunityBot
- 1
0
votes
0
answers
153
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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,\...
CommunityBot
- 1
2
votes
0
answers
134
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Even cycle constrained edge coloring
Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ ...
- 13.9k
1
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1
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175
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Distribution of Induced Subgraphs of Extremal Ramsey Graphs
Choose $k$. Let $G = (V,E)$ be a graph on $n = R(k,k)-1$ vertices (that is, $G$ is an extremal example for $R(k,k)$, and $g : E \to \{r, b\}$ be an edge 2-coloring such that there is no monochromatic $...
- 686
2
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1
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482
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Suppose the independent number of a graph is bounded. How small the clique number can be?
Suppose the independent number of a graph is bounded. How small the clique number can be? linear?
It seems to be a natural problem to ask. but I could not find any reference.
Thanks.
- 82.7k