scholarly journals Multicolor Ramsey Numbers via Pseudorandom Graphs

10.37236/9071 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Xiaoyu He ◽  
Yuval Wigderson
Keyword(s):  
Spectral Expansion ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Positive Integers

A weakly optimal $K_s$-free $(n,d,\lambda)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=\Theta(n^{1-\alpha})$ and spectral expansion $\lambda=\Theta(n^{1-(s-1)\alpha})$, for some fixed $\alpha>0$. Such a graph is called optimal if additionally $\alpha = \frac{1}{2s-3}$. We prove that if $s_{1},\ldots,s_{k}\ge3$ are fixed positive integers and weakly optimal $K_{s_{i}}$-free pseudorandom graphs exist for each $1\le i\le k$, then the multicolor Ramsey numbers satisfy\[\Omega\Big(\frac{t^{S+1}}{\log^{2S}t}\Big)\le r(s_{1},\ldots,s_{k},t)\le O\Big(\frac{t^{S+1}}{\log^{S}t}\Big),\]as $t\rightarrow\infty$, where $S=\sum_{i=1}^{k}(s_{i}-2)$. This generalizes previous results of Mubayi and Verstra\"ete, who proved the case $k=1$, and Alon and Rödl, who proved the case $s_1=\cdots = s_k = 3$. Both previous results used the existence of optimal rather than weakly optimal $K_{s_i}$-free graphs.

Mixed Ramsey Numbers Revisited

2003 ◽  
Vol 12 (5-6) ◽  
pp. 653-660
Author(s):  
C. C. Rousseau ◽  
S. E. Speed
Keyword(s):  
Asymptotic Behaviour ◽  
Ramsey Number ◽  
Open Problems ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Dense Graphs

Given a graph Hwith no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with H fixed and , (ii) with m fixed and a sequence of dense graphs, in particular for the sequence . Open problems are mentioned throughout the paper.

scholarly journals Graphs on Affine and Linear Spaces and Deuber Sets

10.37236/2732 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
David S. Gunderson ◽  
Hanno Lefmann
Keyword(s):  
Abelian Group ◽  
Finite Field ◽  
Recent Work ◽  
Independent Set ◽  
Large Set ◽  
Free Graph ◽  
Linear Spaces ◽  
Ramsey’S Theorem ◽  
Positive Integers ◽  
Arbitrary Abelian Group

If $G$ is a large $K_k$-free graph, by Ramsey's theorem, a large set of vertices is independent. For graphs whose vertices are positive integers, much recent work has been done to identify what arithmetic structure is possible in an independent set. This paper addresses  similar problems: for graphs whose vertices are affine or linear spaces over a finite field,  and when the vertices of the graph are elements of an arbitrary Abelian group.

scholarly journals New Computational Upper Bounds for Ramsey Numbers $R(3,k)$

10.37236/2824 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jan Goedgebeur ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bounds ◽  
Independent Sets ◽  
Upper Bounds ◽  
Computational Techniques ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Critical Graph ◽  
Minimum Number

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \le 42$, $R(3,11) \le 50$, $R(3,13) \le 68$, $R(3,14) \le 77$, $R(3,15) \le 87$, and $R(3,16) \le 98$. All of them are improvements by one over the previously best known bounds. Let $e(3,k,n)$ denote the minimum number of edges in any triangle-free graph on $n$ vertices without independent sets of order $k$. The new upper bounds on $R(3,k)$ are obtained by completing the computation of the exact values of $e(3,k,n)$ for all $n$ with $k \leq 9$ and for all $n \leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \le k \le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \le 9$. We prove that the known critical graph for $R(3,9)$ on 35 vertices is unique up to isomorphism. For the case of $R(3,10)$, first we establish that $R(3,10)=43$ if and only if $e(3,10,42)=189$, or equivalently, that if $R(3,10)=43$ then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that $R(3,10) \le 42$.

scholarly journals The complexity of frugal colouring

2021 ◽  
Author(s):  
Stefan Bard ◽  
Gary MacGillivray ◽  
Shayla Redlin
Keyword(s):  
Positive Integer ◽  
Maximum Degree ◽  
Free Graph ◽  
Dichotomy Theorem ◽  
Graph Classes ◽  
Positive Integers ◽  
Thomas Theorem

AbstractA t-frugal colouring of a graph G is an assignment of colours to the vertices of G, such that each colour appears at most t times in the neighbourhood of any vertex. A dichotomy theorem for the complexity of deciding whether a graph has a 1-frugal colouring with k colours was found by McCormick and Thomas, and then later extended to restricted graph classes by Kratochvil and Siggers. We generalize the McCormick and Thomas theorem by proving a dichotomy theorem for the complexity of deciding whether a graph has a t-frugal colouring with k colours, for all pairs of positive integers t and k. We also generalize bounds of Lih et al. for the number of colours needed in a 1-frugal colouring of a given $$K_4$$ K 4 -minor-free graph with maximum degree $$\Delta $$ Δ to t-frugal colourings, for any positive integer t.

scholarly journals Generalizing the Ramsey Problem through Diameter

10.37236/1658 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Dhruv Mubayi
Keyword(s):  
Prime Power ◽  
Ramsey Problem ◽  
Ramsey Numbers ◽  
The Third ◽  
Positive Integers

Given a graph $G$ and positive integers $d,k$, let $f_d^k(G)$ be the maximum $t$ such that every $k$-coloring of $E(G)$ yields a monochromatic subgraph with diameter at most $d$ on at least $t$ vertices. Determining $f_1^k(K_n)$ is equivalent to determining classical Ramsey numbers for multicolorings. Our results include $\bullet$ determining $f_d^k(K_{a,b})$ within 1 for all $d,k,a,b$ $\bullet$ for $d \ge 4$, $f_d^3(K_n)=\lceil n/2 \rceil +1$ or $\lceil n/2 \rceil$ depending on whether $n \equiv 2 (mod 4)$ or not $\bullet$ $f_3^k(K_n) > {{n}\over {k-1+1/k}}$ The third result is almost sharp, since a construction due to Calkin implies that $f_3^k(K_n) \le {{n}\over {k-1}} +k-1$ when $k-1$ is a prime power. The asymptotics for $f_d^k(K_n)$ remain open when $d=k=3$ and when $d\ge 3, k \ge 4$ are fixed.

scholarly journals Three Color Ramsey Numbers for Graphs with at most 4 Vertices

10.37236/2160 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Luis Boza ◽  
Janusz Dybizbański ◽  
Tomasz Dzido
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Monochromatic Copy

For given graphs $H_{1}, H_{2}, H_{3}$, the 3-color Ramsey number $R(H_{1},$ $H_{2}, H_{3})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $3$ colors, then it always contains a monochromatic copy of $H_{i}$ colored with $i$, for some $1 \leq i \leq 3$.We study the bounds on 3-color Ramsey numbers $R(H_1,H_2,H_3)$, where $H_i$ is an isolate-free graph different from $K_2$ with at most four vertices, establishing that $R(P_4,C_4,K_4)=14$, $R(C_4,K_3,K_4-e)=17$, $R(C_4,K_3+e,K_4-e)=17$, $R(C_4,K_4-e,K_4-e)=19$, $28\le R(C_4,K_4-e,K_4)\le36$, $R(K_3,K_4-e,K_4)\le41$, $R(K_4-e,K_4-e,K_4)\le59$ and $R(K_4-e,K_4,K_4)\le113$. Also, we prove that $R(K_3+e,K_4-e,K_4-e)=R(K_3,K_4-e,K_4-e)$, $R(C_4,K_3+e,K_4)\le\max\{R(C_4,K_3,K_4),29\}\le32$, $R(K_3+e,K_4-e,K_4)\le\max\{R(K_3,K_4-e,K_4),33\}\le41$ and $R(K_3+e,K_4,K_4)\le\max\{R(K_3,K_4,K_4),2R(K_3,K_3,K_4)+2\}\le79$.This paper is an extension of the article by Arste, Klamroth, Mengersen [Utilitas Mathematica, 1996].

Some Small Circuit-Cocircuit Ramsey Numbers for Matroids

1995 ◽  
Vol 4 (1) ◽  
pp. 67-80 ◽  
Author(s):  
Fair Barbour Hurst ◽  
Talmage James Reid
Keyword(s):  
Positive Integer ◽  
Matroid Theory ◽  
Ramsey Numbers ◽  
Connected Matroid ◽  
Positive Integers

Ramsey numbers for matroids, which mimic properties of Ramsey numbers for graphs, have been denned as follows. Let k and l be positive integers. Then n(k, l) is the least positive integer n such that every connected matroid with n elements contains either a circuit with at least k elements or a cocircuit with at least l elements. We determine the largest known value of these numbers in the sense of maximizing both k and l. We also find extremal matroids with small circuits and cocircuits. Results on matroid connectivity, geometry, and extremal matroid theory are used here.

scholarly journals The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$

10.37236/3684 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Jan Goedgebeur ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Ramsey Number ◽  
Computer Algorithms ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Open Case ◽  
Minimum Number ◽  
Graph Generation

Using computer algorithms we establish that the Ramsey number $R(3,K_{10}-e)$ is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of $R(3,K_k-e)$ for $11 \le k \le 16$, and show by construction a new lower bound $55 \le R(3,K_{13}-e)$.The new upper bounds on $R(3,K_k-e)$ are obtained by using the values and lower bounds on $e(3,K_l-e,n)$ for $l \le k$, where $e(3,K_k-e,n)$ is the minimum number of edges in any triangle-free graph on $n$ vertices without $K_k-e$ in the complement. We complete the computation of the exact values of $e(3,K_k-e,n)$ for all $n$ with $k \leq 10$ and for $n \leq 34$ with $k = 11$, and establish many new lower bounds on $e(3,K_k-e,n)$ for higher values of $k$.Using the maximum triangle-free graph generation method, we determine two other previously unknown Ramsey numbers, namely $R(3,K_{10}-K_3-e)=31$ and $R(3,K_{10}-P_3-e)=31$. For graphs $G$ on 10 vertices, besides $G=K_{10}$, this leaves 6 open cases of the form $R(3,G)$. The hardest among them appears to be $G=K_{10}-2K_2$, for which we establish the bounds $31 \le R(3,K_{10}-2K_2) \le 33$.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

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