14,008 research outputs found
What is Ramsey-equivalent to a clique?
A graph G is Ramsey for H if every two-colouring of the edges of G contains a
monochromatic copy of H. Two graphs H and H' are Ramsey-equivalent if every
graph G is Ramsey for H if and only if it is Ramsey for H'. In this paper, we
study the problem of determining which graphs are Ramsey-equivalent to the
complete graph K_k. A famous theorem of Nesetril and Rodl implies that any
graph H which is Ramsey-equivalent to K_k must contain K_k. We prove that the
only connected graph which is Ramsey-equivalent to K_k is itself. This gives a
negative answer to the question of Szabo, Zumstein, and Zurcher on whether K_k
is Ramsey-equivalent to K_k.K_2, the graph on k+1 vertices consisting of K_k
with a pendent edge.
In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph
H if it is Ramsey for H but no proper subgraph of G is Ramsey for H. Let s(H)
be the smallest minimum degree over all Ramsey minimal graphs for H. The study
of s(H) was introduced by Burr, Erdos, and Lovasz, where they show that
s(K_k)=(k-1)^2. We prove that s(K_k.K_2)=k-1, and hence K_k and K_k.K_2 are not
Ramsey-equivalent.
We also address the question of which non-connected graphs are
Ramsey-equivalent to K_k. Let f(k,t) be the maximum f such that the graph
H=K_k+fK_t, consisting of K_k and f disjoint copies of K_t, is
Ramsey-equivalent to K_k. Szabo, Zumstein, and Zurcher gave a lower bound on
f(k,t). We prove an upper bound on f(k,t) which is roughly within a factor 2 of
the lower bound
Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques
For graphs and , we say is Ramsey for if every -coloring of
the edges of contains a monochromatic copy of . The graph is Ramsey
-minimal if is Ramsey for and there is no proper subgraph of
so that is Ramsey for . Burr, Erdos, and Lovasz defined to
be the minimum degree of over all Ramsey -minimal graphs . Define
to be a graph on vertices consisting of a complete graph on
vertices and one additional vertex of degree . We show that
for all values ; it was previously known that , so it
is surprising that is much smaller.
We also make some further progress on some sparser graphs. Fox and Lin
observed that for all graphs , where is
the minimum degree of ; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs without
isolated vertices satisfy . Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that -regular -connected
triangle-free graphs , with one extra technical constraint, satisfy ; the extra constraint is that has a vertex so that if one
removes and its neighborhood from , the remainder is connected.Comment: 10 pages; 3 figure
BEBERAPA KELAS GRAF RAMSEY MINIMAL UNTUK LINTASAN P_3 VERSUS P_5
In 1930, Frank Plumpton Ramsey has introduced Ramsey's theory, in his paper titled On a Problem of Formal Logic. This study became morepopular since Erdős and Szekeres applied Ramsey's theory to graph theory. Suppose given the graph F, G and H. The notation F → (G, H) states thatfor any red-blue coloring of the edges of F implies F containing a red subgraph of G or a blue subgraph of H. The graph F is said to be the Ramsey graph for graph G versus H (pair G and H) if F → (G, H). Graph F is called Ramsey minimal graph for G versus H if first, F → (G, H) and second, F satisfies the minimality property i.e. for each e ∈ E (F), then F-e ↛ (G, H). The class of all Ramsey (G, H) minimal graphs is denoted by (G, H). The class (G, H) is called Ramsey infinite or finite if (G, H) is infinite or finite, respectively. The study about Ramsey minimal graph is still continuously being developed and examined, although in general it is not easy to characterize or determine the graphs included in the (G, H), especially if (G, H) is an infinite Ramsey class. The characterization of graphs in (, ) has been obtained. However, the characterization of graphs in (, ), for every 3 ≤ m < n is still open. In this article, we will determine some infinite classes of Ramsey minimal graphs for paths versus .
On the minimum degree of minimal Ramsey graphs for multiple colours
A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every
r-colouring of the edges of G contains a monochromatic copy of H. The graph G
is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph
of G possesses this property. Let s_r(H) denote the smallest minimum degree of
G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter
s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed
that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of
s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) =
r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in
both r and k, and we determine s_r(K_3) up to a factor of log r
Ramsey expansions of metrically homogeneous graphs
We discuss the Ramsey property, the existence of a stationary independence
relation and the coherent extension property for partial isometries (coherent
EPPA) for all classes of metrically homogeneous graphs from Cherlin's
catalogue, which is conjectured to include all such structures. We show that,
with the exception of tree-like graphs, all metric spaces in the catalogue have
precompact Ramsey expansions (or lifts) with the expansion property. With two
exceptions we can also characterise the existence of a stationary independence
relation and the coherent EPPA.
Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's
classification programme of Ramsey classes and as empirical evidence of the
recent convergence in techniques employed to establish the Ramsey property, the
expansion (or lift or ordering) property, EPPA and the existence of a
stationary independence relation. At the heart of our proof is a canonical way
of completing edge-labelled graphs to metric spaces in Cherlin's classes. The
existence of such a "completion algorithm" then allows us to apply several
strong results in the areas that imply EPPA and respectively the Ramsey
property.
The main results have numerous corollaries on the automorphism groups of the
Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity,
existence of universal minimal flows, ample generics, small index property,
21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor
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