Ramsey's theorem
- Task
Find a graph with 17 Nodes such that any 4 Nodes are neither totally connected nor totally unconnected, so demonstrating Ramsey's theorem.
A specially-nominated solution may be used, but if so it must be checked to see if if there are any sub-graphs that are totally connected or totally unconnected.
C
For 17 nodes, (4,4) happens to have a special solution: arrange nodes on a circle, and connect all pairs with distances 1, 2, 4, and 8. It's easier to prove it on paper and just show the result than let a computer find it (you can call it optimization). <lang c>#include <stdio.h>
int a[17][17], idx[4];
int find_group(int type, int min_n, int max_n, int depth) { int i, n; if (depth == 4) { printf("totally %sconnected group:", type ? "" : "un"); for (i = 0; i < 4; i++) printf(" %d", idx[i]); putchar('\n'); return 1; }
for (i = min_n; i < max_n; i++) { for (n = 0; n < depth; n++) if (a[idx[n]][i] != type) break;
if (n == depth) { idx[n] = i; if (find_group(type, 1, max_n, depth + 1)) return 1; } } return 0; }
int main() { int i, j, k; const char *mark = "01-";
for (i = 0; i < 17; i++) a[i][i] = 2;
for (k = 1; k <= 8; k <<= 1) { for (i = 0; i < 17; i++) { j = (i + k) % 17; a[i][j] = a[j][i] = 1; } }
for (i = 0; i < 17; i++) { for (j = 0; j < 17; j++) printf("%c ", mark[a[i][j]]); putchar('\n'); }
// testcase breakage // a[2][1] = a[1][2] = 0;
// it's symmetric, so only need to test groups containing node 0 for (i = 0; i < 17; i++) { idx[0] = i; if (find_group(1, i+1, 17, 1) || find_group(0, i+1, 17, 1)) { puts("no good"); return 0; } } puts("all good"); return 0; }</lang>
- Output:
(17 x 17 connectivity matrix)
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - all good
D
<lang d>import std.stdio, std.string, std.algorithm, std.range;
/// Generate the connectivity matrix. immutable(char)[][] generateMatrix() {
immutable r = format("-%b", 53643);
return r.length.iota.map!(i => r[$-i .. $] ~ r[0 .. $-i]).array;
}
/**Check that every clique of four has at least one pair connected and one pair unconnected. It requires a symmetric matrix.*/ string ramseyCheck(in char[][] mat) pure @safe in {
foreach (immutable r, const row; mat) {
assert(row.length == mat.length);
foreach (immutable c, immutable x; row)
assert(x == mat[c][r]);
}
} body {
immutable N = mat.length; char[6] connectivity = '-';
foreach (immutable a; 0 .. N) {
foreach (immutable b; 0 .. N) {
if (a == b) continue;
connectivity[0] = mat[a][b];
foreach (immutable c; 0 .. N) {
if (a == c || b == c) continue;
connectivity[1] = mat[a][c];
connectivity[2] = mat[b][c];
foreach (immutable d; 0 .. N) {
if (a == d || b == d || c == d) continue;
connectivity[3] = mat[a][d];
connectivity[4] = mat[b][d];
connectivity[5] = mat[c][d];
// We've extracted a meaningful subgraph,
// check its connectivity.
if (!connectivity[].canFind('0'))
return format("Fail, found wholly connected: ",
a, " ", b," ", c, " ", d);
else if (!connectivity[].canFind('1'))
return format("Fail, found wholly " ~
"unconnected: ",
a, " ", b," ", c, " ", d);
}
}
}
}
return "Satisfies Ramsey condition.";
}
void main() {
const mat = generateMatrix;
writefln("%-(%(%c %)\n%)", mat);
mat.ramseyCheck.writeln;
}</lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - Satisfies Ramsey condition.
Elixir
<lang elixir>defmodule Ramsey do
def main(n\\17) do
vertices = Enum.to_list(0 .. n-1)
g = create_graph(n,vertices)
edges = for v1 <- :digraph.vertices(g), v2 <- :digraph.out_neighbours(g, v1), do: {v1,v2}
print_graph(vertices,edges)
case ramsey_check(vertices,edges) do
true -> "Satisfies Ramsey condition."
{false,reason} -> "Not satisfies Ramsey condition:\n#{inspect reason}"
end
|> IO.puts
end
def create_graph(n,vertices) do
g = :digraph.new([:cyclic])
for v <- vertices, do: :digraph.add_vertex(g,v)
for i <- vertices, k <- [1,2,4,8] do
j = rem(i + k, n)
:digraph.add_edge(g, i, j)
:digraph.add_edge(g, j, i)
end
g
end
def print_graph(vertices,edges) do
Enum.each(vertices, fn j ->
Enum.map_join(vertices, " ", fn i ->
cond do
i==j -> "-"
{i,j} in edges -> "1"
true -> "0"
end
end)
|> IO.puts
end)
end
def ramsey_check(vertices,edges) do
listconditions =
for v1 <- vertices, v2 <- vertices, v3 <- vertices, v4 <- vertices,
v1 != v2, v1 != v3, v1 != v4, v2 != v3, v2 != v4, v3 != v4
do
all_cases = [ {v1,v2} in edges, {v1,v3} in edges, {v1,v4} in edges,
{v2,v3} in edges, {v2,v4} in edges, {v3,v4} in edges ]
{v1, v2, v3, v4, Enum.any?(all_cases), not(Enum.all?(all_cases))}
end
if Enum.all?(listconditions, fn {_,_,_,_,c1,c2} -> c1 and c2 end) do
true
else
{false, (for {v1,v2,v3,v4,false,_} <- listconditions, do: {:wholly_unconnected,v1,v2,v3,v4})
++ (for {v1,v2,v3,v4,_,false} <- listconditions, do: {:wholly_connected,v1,v2,v3,v4}) }
end
end
end
Ramsey.main</lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - Satisfies Ramsey condition.
Erlang
<lang erlang>-module(ramsey_theorem). -export([main/0]).
main() -> Vertices = lists:seq(0,16), G = create_graph(Vertices), String_ramsey = case ramsey_check(G,Vertices) of true -> "Satisfies Ramsey condition."; {false,Reason} -> "Not satisfies Ramsey condition:\n" ++ io_lib:format("~p\n",[Reason]) end, io:format("~s\n~s\n",[print_graph(G,Vertices),String_ramsey]).
create_graph(Vertices) -> G = digraph:new([cyclic]), [digraph:add_vertex(G,V) || V <- Vertices], [begin J = ((I + K) rem 17), digraph:add_edge(G, I, J), digraph:add_edge(G, J, I) end || I <- Vertices, K <- [1,2,4,8]], G.
print_graph(G,Vertices) -> Edges = [{V1,V2} || V1 <- digraph:vertices(G), V2 <- digraph:out_neighbours(G, V1)], lists:flatten( [[ [case I of J -> $-; _ -> case lists:member({I,J},Edges) of true -> $1; false -> $0 end end,$ ] || I <- Vertices] ++ [$\n] || J <- Vertices]).
ramsey_check(G,Vertices) -> Edges = [{V1,V2} || V1 <- digraph:vertices(G), V2 <- digraph:out_neighbours(G, V1)], ListConditions = [begin All_cases = [lists:member({V1,V2},Edges), lists:member({V1,V3},Edges), lists:member({V1,V4},Edges), lists:member({V2,V3},Edges), lists:member({V2,V4},Edges), lists:member({V3,V4},Edges)], {V1,V2,V3,V4, lists:any(fun(X) -> X end, All_cases), not(lists:all(fun(X) -> X end, All_cases))} end || V1 <- Vertices, V2 <- Vertices, V3 <- Vertices, V4 <- Vertices, V1/=V2,V1/=V3,V1/=V4,V2/=V3,V2/=V4,V3/=V4], case lists:all(fun({_,_,_,_,C1,C2}) -> C1 and C2 end,ListConditions) of true -> true; false -> {false, [{wholly_unconnected,V1,V2,V3,V4} || {V1,V2,V3,V4,false,_} <- ListConditions] ++ [{wholly_connected,V1,V2,V3,V4} || {V1,V2,V3,V4,_,false} <- ListConditions]} end.</lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - Satisfies Ramsey condition.
J
Interpreting this task as "reproduce the output of all the other examples", then here's a stroll to the goal through the J interpreter: <lang j> i.@<.&.(2&^.) N =: 17 NB. Count to N by powers of 2 1 2 4 8
1 #~ 1 j. 0 _1:} i.@<.&.(2&^.) N =: 17 NB. Turn indices into bit mask
1 0 1 0 0 1 0 0 0 0 1
(, |.) 1 #~ 1 j. 0 _1:} i.@<.&.(2&^.) N =: 17 NB. Cat the bitmask with its own reflection
1 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1
_1 |.^:(<N) _ , (, |.) 1 #~ 1 j. 0 _1:} <: i.@<.&.(2&^.) N=:17 NB. Then rotate N times to produce the array
_ 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 _
NB. Packaged up as a re-usable function ramsey =: _1&|.^:((<@])`(_ , [: (, |.) 1 #~ 1 j. 0 _1:} [: <: i.@<.&.(2&^.)@])) ramsey 17
_ 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 _ 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 _</lang>
To test if all combinations of 4 rows and columns contain both a 0 and a 1 <lang j>
comb=: 4 : 0 M. NB. All size x combinations of i.y
if. (x>:y)+.0=x do. i.(x<:y),x else. (0,.x comb&.<: y),1+x comb y-1 end.
)
NB. returns 1 iff the subbmatrix of y consisting of the columns and rows labelled x contains both 1 and 0
checkRow =. 4 : 0 "1 _
*./ 0 1 e. ,x{"1 x{y
)
*./ (4 comb 17) checkRow ramsey 17
1 </lang>
Java
Translation of Tcl via D
<lang java>import java.util.Arrays; import java.util.stream.IntStream;
public class RamseysTheorem {
static char[][] createMatrix() {
String r = "-" + Integer.toBinaryString(53643);
int len = r.length();
return IntStream.range(0, len)
.mapToObj(i -> r.substring(len - i) + r.substring(0, len - i))
.map(String::toCharArray)
.toArray(char[][]::new);
}
/**
* Check that every clique of four has at least one pair connected and one
* pair unconnected. It requires a symmetric matrix.
*/
static String ramseyCheck(char[][] mat) {
int len = mat.length;
char[] connectivity = "------".toCharArray();
for (int a = 0; a < len; a++) {
for (int b = 0; b < len; b++) {
if (a == b)
continue;
connectivity[0] = mat[a][b];
for (int c = 0; c < len; c++) {
if (a == c || b == c)
continue;
connectivity[1] = mat[a][c];
connectivity[2] = mat[b][c];
for (int d = 0; d < len; d++) {
if (a == d || b == d || c == d)
continue;
connectivity[3] = mat[a][d];
connectivity[4] = mat[b][d];
connectivity[5] = mat[c][d];
// We've extracted a meaningful subgraph,
// check its connectivity.
String conn = new String(connectivity);
if (conn.indexOf('0') == -1)
return String.format("Fail, found wholly connected: "
+ "%d %d %d %d", a, b, c, d);
else if (conn.indexOf('1') == -1)
return String.format("Fail, found wholly unconnected: "
+ "%d %d %d %d", a, b, c, d);
}
}
}
}
return "Satisfies Ramsey condition.";
}
public static void main(String[] a) {
char[][] mat = createMatrix();
for (char[] s : mat)
System.out.println(Arrays.toString(s));
System.out.println(ramseyCheck(mat));
}
}</lang>
[-, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1] [1, -, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1] [1, 1, -, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0] [0, 1, 1, -, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1] [1, 0, 1, 1, -, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0] [0, 1, 0, 1, 1, -, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0] [0, 0, 1, 0, 1, 1, -, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0] [0, 0, 0, 1, 0, 1, 1, -, 1, 1, 0, 1, 0, 0, 0, 1, 1] [1, 0, 0, 0, 1, 0, 1, 1, -, 1, 1, 0, 1, 0, 0, 0, 1] [1, 1, 0, 0, 0, 1, 0, 1, 1, -, 1, 1, 0, 1, 0, 0, 0] [0, 1, 1, 0, 0, 0, 1, 0, 1, 1, -, 1, 1, 0, 1, 0, 0] [0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, -, 1, 1, 0, 1, 0] [0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, -, 1, 1, 0, 1] [1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, -, 1, 1, 0] [0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, -, 1, 1] [1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, -, 1] [1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, -] Satisfies Ramsey condition.
Kotlin
<lang scala>// version 1.1.0
val a = Array(17) { IntArray(17) } val idx = IntArray(4)
fun findGroup(type: Int, minN: Int, maxN: Int, depth: Int): Boolean {
if (depth == 4) {
print("\nTotally ${if (type != 0) "" else "un"}connected group:")
for (i in 0 until 4) print(" ${idx[i]}")
println()
return true
}
for (i in minN until maxN) {
var n = depth
for (m in 0 until depth) if (a[idx[m]][i] != type) {
n = m
break
}
if (n == depth) {
idx[n] = i
if (findGroup(type, 1, maxN, depth + 1)) return true
}
}
return false
}
fun main(args: Array<String>) {
for (i in 0 until 17) a[i][i] = 2
var j: Int
var k = 1
while (k <= 8) {
for (i in 0 until 17) {
j = (i + k) % 17
a[i][j] = 1
a[j][i] = 1
}
k = k shl 1
}
val mark = "01-"
for (i in 0 until 17) {
for (m in 0 until 17) print("${mark[a[i][m]]} ")
println()
}
for (i in 0 until 17) {
idx[0] = i
if (findGroup(1, i + 1, 17, 1) || findGroup(0, i + 1, 17, 1)) {
println("\nRamsey condition not satisfied.")
return
}
}
println("\nRamsey condition satisfied.")
}</lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - Ramsey condition satisfied.
Mathematica
<lang mathematica>CirculantGraph[17, {1, 2, 4, 8}]</lang>
Mathprog
<lang>/*Ramsey 4 4 17
This model finds a graph with 17 Nodes such that no clique of 4 Nodes is either fully connected, nor fully disconnected Nigel_Galloway January 18th., 2012
- /
param Nodes := 17; var Arc{1..Nodes, 1..Nodes}, binary;
clique{a in 1..(Nodes-3), b in (a+1)..(Nodes-2), c in (b+1)..(Nodes-1), d in (c+1)..Nodes} : 1 <= Arc[a,b] + Arc[a,c] + Arc[a,d] + Arc[b,c] + Arc[b,d] + Arc[c,d] <= 5;
end;</lang>
This may be run with: <lang bash>glpsol --minisat --math R_4_4_17.mprog --output R_4_4_17.sol</lang> The solution may be viewed on this page. In the solution file, the first section identifies the number of nodes connected in this clique. In the second part of the solution, the status of each arc in the graph (connected=1, unconnected=0) is shown.
PARI/GP
This takes the C solution to its logical extreme. <lang parigp>
check(M)={
my(n=#M);
for(a=1,n-3,
for(b=a+1,n-2,
my(goal=!M[a,b]);
for(c=b+1,n-1,
if(M[a,c]==goal || M[b,c]==goal, next(2));
for(d=c+1,n,
if(M[a,d]==goal || M[b,d]==goal || M[c,d]==goal, next(3));
)
);
print(a" "b);
return(0)
)
);
1
};
M=matrix(17,17,x,y,my(t=abs(x-y)%17);t==2^min(valuation(t,2),3)) check(M)</lang>
Perl 6
<lang perl6>my @a = [ 0 xx 17 ] xx 17; @a[$_;$_] = '-' for ^17;
for flat ^17 X 1,2,4,8 -> $i, $k {
my $j = ($i + $k) % 17; @a[$i;$j] = @a[$j;$i] = 1;
} .say for @a;
for combinations(17,4).Array -> $quartet {
my $links = [+] $quartet.combinations(2).map: -> $i,$j { @a[$i;$j] }
die "Bogus!" unless 0 < $links < 6;
} say "OK";</lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - OK
Python
<lang python>range17 = range(17) a = [['0'] * 17 for i in range17] idx = [0] * 4
def find_group(mark, min_n, max_n, depth=1):
if (depth == 4):
prefix = "" if (mark == '1') else "un"
print("Fail, found totally {}connected group:".format(prefix))
for i in range(4):
print(idx[i])
return True
for i in range(min_n, max_n):
n = 0
while (n < depth):
if (a[idx[n]][i] != mark):
break
n += 1
if (n == depth):
idx[n] = i
if (find_group(mark, 1, max_n, depth + 1)):
return True
return False
if __name__ == '__main__':
for i in range17:
a[i][i] = '-'
for k in range(4):
for i in range17:
j = (i + pow(2, k)) % 17
a[i][j] = a[j][i] = '1'
# testcase breakage # a[2][1] = a[1][2] = '0'
for row in a:
print(' '.join(row))
for i in range17:
idx[0] = i
if (find_group('1', i + 1, 17) or find_group('0', i + 1, 17)):
print("no good")
exit()
print("all good")</lang>
- Output same as C:
Racket
Kind of a translation of C (ie, reducing this problem to generating a printout of a specific matrix). <lang racket>#lang racket
(define N 17)
(define (dist i j)
(define d (abs (- i j))) (if (<= d (quotient N 2)) d (- N d)))
(define v
(build-vector N
(λ(i) (build-vector N
(λ(j) (case (dist i j) [(0) '-] [(1 2 4 8) 1] [else 0]))))))
(for ([row v]) (displayln row))</lang>
REXX
Mainline programming was borrowed from C. <lang rexx>/*REXX program finds and displays a 17 node graph such that any four nodes are neither */ /*─────────────────────────────────────────── totally connected nor totally unconnected.*/ @.=0; #=17 /*initialize the node graph to zero. */
do d=0 for #; @.d.d=2; end /*d*/ /*set the diagonal elements to two. */
do k=1 by 0 while k<=8 /*K is doubled each time through loop.*/
do i=0 for # /*process each column in the array. */
j= (i+k) // # /*set a row,column and column,row. */
@.i.j=1; @.j.i=1 /*set two array elements to unity. */
end /*i*/
k=k+k /*double the value of K for each loop*/
end /*while k≤8*/
/* [↓] display a connection grid. */
do r=0 for #; _=; do c=0 for # /*show each row, build column by column*/
_=_ @.r.c /*add (append) the column to the row.*/
end /*c*/
say left(, 9) translate(_, "-", 2) /*display the constructed row. */
end /*r*/
/*verify the sub-graphs connections. */
!.=0; ok=1 /*Ramsey's connections; OK (so far).*/
/* [↓] check col. with row connections*/
do v=0 for # /*check sub-graphs number connections. */
do h=0 for # /*check column connections to the rows.*/
if @.v.h==1 then !._v.v=!._v.v+1 /*if connected, then bump the counter.*/
end /*h*/ /* [↑] Note: we're counting each */
ok=ok & !._v.v==#%2 /* connection twice, so divide */
end /*v*/ /* the total by two. */
/* [↓] check col. with row connections*/
do h=0 for # /*check the sub-graphs # of connections*/
do v=0 for # /*check the row connection to a column.*/
if @.h.v==1 then !._h.h=!._h.h+1 /*if connected, then bump the counter.*/
end /*v*/ /* [↑] Note: we're counting each */
ok=ok & !._h.h==#%2 /* connection twice, so divide */
end /*h*/ /* the total by two. */
say /* [↓] a yea─or─nay message. */ say space("Ramsey's condition" word('not', 1+ok) "satisfied.")
/*stick a fork in it, we're all done. */</lang>
output (17x17 connectivity matrix):
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1
1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1
1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0
0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1
1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0
0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0
0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0
0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1
1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1
1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0
0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0
0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0
0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1
1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0
0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1
1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1
1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 -
Ramsey's condition satisfied.
Ruby
<lang ruby>a = Array.new(17){['0'] * 17} 17.times{|i| a[i][i] = '-'} 4.times do |k|
17.times do |i| j = (i + 2 ** k) % 17 a[i][j] = a[j][i] = '1' end
end a.each {|row| puts row.join(' ')}
- check taken from Perl6 version
(0...17).to_a.combination(4) do |quartet|
links = quartet.combination(2).map{|i,j| a[i][j].to_i}.reduce(:+)
abort "Bogus" unless 0 < links && links < 6
end puts "Ok" </lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - Ok
Run BASIC
<lang runbasic>dim a(17,17) for i = 1 to 17: a(i,i) = -1: next i k = 1 while k <= 8
for i = 1 to 17 j = (i + k) mod 17 a(i,j) = 1 a(j,i) = 1 next i k = k * 2
wend for i = 1 to 17
for j = 1 to 17 print a(i,j);" "; next j print
next i</lang>
-1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 -1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 -1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 -1 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 -1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 -1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 -1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 -1 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 -1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 -1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 -1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 -1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 -1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 -1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 -1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 -1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 -1
Sidef
<lang ruby>var a = 17.of { 17.of(0) }
17.itimes{|i| a[i][i] = '-' } 4.itimes { |k|
17.itimes { |i|
var j = ((i + 1<<k) % 17)
a[i][j] = (a[j][i] = 1)
}
}
a.each {|row| say row.join(' ') }
@(0..16).combinations(4).each { |quartet|
var links = quartet.combinations(2).map{|p| a.dig(p...) }.sum
((0 < links) && (links < 6)) || die "Bogus!"
} say "Ok"</lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - Ok
Tcl
<lang tcl>package require Tcl 8.6
- Generate the connectivity matrix
set init [split [format -%b 53643] ""] set matrix {} for {set r $init} {$r ni $matrix} {set r [concat [lindex $r end] [lrange $r 0 end-1]]} {
lappend matrix $r
}
- Check that every clique of four has at least *one* pair connected and one
- pair unconnected. ASSUMES that the graph is symmetric.
proc ramseyCheck4 {matrix} {
set N [llength $matrix]
set connectivity [lrepeat 6 -]
for {set a 0} {$a < $N} {incr a} {
for {set b 0} {$b < $N} {incr b} { if {$a==$b} continue lset connectivity 0 [lindex $matrix $a $b] for {set c 0} {$c < $N} {incr c} { if {$a==$c || $b==$c} continue lset connectivity 1 [lindex $matrix $a $c] lset connectivity 2 [lindex $matrix $b $c] for {set d 0} {$d < $N} {incr d} { if {$a==$d || $b==$d || $c==$d} continue lset connectivity 3 [lindex $matrix $a $d] lset connectivity 4 [lindex $matrix $b $d] lset connectivity 5 [lindex $matrix $c $d]
# We've extracted a meaningful subgraph; check its connectivity if {0 ni $connectivity} { puts "FAIL! Found wholly connected: $a $b $c $d" return } elseif {1 ni $connectivity} { puts "FAIL! Found wholly unconnected: $a $b $c $d" return } } } }
} puts "Satisfies Ramsey condition"
}
puts [join $matrix \n] ramseyCheck4 $matrix</lang>
- Output:
- 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 - Satisfies Ramsey condition