The party problem: R(3,3) = 6, proved rather than quoted

Among any six people, three are mutual acquaintances or three are mutual strangers; five is not enough. The upper half follows from Goodman's identity, which forces at least two monochromatic triangles at \(n = 6\). The lower half needs a witness, and the five-cycle colouring is one – acquaintance around a pentagon, strangers on the diagonals, and neither the pentagon nor the pentagram contains a triangle.

Usage

morie_party_problem(n_people = 6L)

Arguments

n_people

Size of the party, at least 3.

Value

A list with guaranteed, minimum_monochromatic, witness, witness_valid, ramsey_number.

References

Goodman AW (1959); Greenwood RE, Gleason AM (1955).

Examples

morie_party_problem()
#> $n_people
#> [1] 6
#> 
#> $guaranteed
#> [1] TRUE
#> 
#> $minimum_monochromatic
#> [1] 2
#> 
#> $witness
#> NULL
#> 
#> $witness_valid
#> NULL
#> 
#> $ramsey_number
#> [1] 6
#> 
#> $n
#> [1] 6
#> 
#> $method
#> [1] "Party problem via Goodman's identity"
#>