Editing Ramsey's theorem (section)

==== More colors ====
We can also generalize the induced Ramsey's theorem to a multicolor setting. For graphs {{math|''H''{{sub|1}}, ''H''{{sub|2}}, …, ''H{{sub|r}}''}}, define {{math|''r''{{sub|ind}}(''H''{{sub|1}}, ''H''{{sub|2}}, …, ''H{{sub|r}}'')}} to be the minimum number of vertices in a graph {{mvar|G}} such that, given any coloring of the edges of {{mvar|G}} into {{mvar|r}} colors, there exists an {{mvar|i}} such that {{math|1 ≤ ''i'' ≤ ''r''}} and such that {{mvar|G}} contains an induced subgraph isomorphic to {{mvar|H{{sub|i}}}} whose edges are all colored in the {{mvar|i}}-th color. Let {{math|1=''r''{{sub|ind}}(''H'';''q'') := ''r''{{sub|ind}}(''H'', ''H'', …, ''H'')}} ({{mvar|q}} copies of {{mvar|H}}).

It is possible to derive a bound on {{math|''r''{{sub|ind}}(''H'';''q'')}} which is approximately a [[Tower of twos|tower of two]] of height {{math|~ log ''q''}} by iteratively applying the bound on the two-color case. The current best known bound is due to Fox and Sudakov, which achieves {{math|''r''{{sub|ind}}(''H'';''q'') ≤ 2{{sup|''ck''{{sup|3}}}}}}, where {{mvar|k}} is the number of vertices of {{mvar|H}} and {{mvar|c}} is a constant depending only on {{mvar|q}}.<ref>{{cite journal
| last1=Fox | first1=Jacob | author-link1=Jacob Fox
| last2=Sudakov | first2=Benny | author-link2=Benny Sudakov
| arxiv=0707.4159v2
| title=Density theorems for bipartite graphs and related Ramsey-type results
| journal=[[Combinatorica]]
| volume=29
| pages=153–196
| year=2009
| issue=2 | doi=10.1007/s00493-009-2475-5 | doi-access=free}}</ref>