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EXAMPLE
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The only 3-uniform Veblen hypergraph with 3 edges is the single edge with multiplicity 3, {(1,2,3)^3}.
The only 3-uniform Veblen hypergraph with 4 edges is the 4-uniform simplex (i.e., the tetrahedron) as shown in Cooper and Dutle.
There are two 3-uniform Veblen hypergraphs with 5 edges: the crown, {(1,2,3),(1,2,4),(1,2,5),(3,4,5)^2}, and the tight 5-cycle, {(1,2,3),(2,3,4),(3,4,5),(4,5,1),(5,1,2)}.
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PROG
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(Sage)
e = n
#Given a 3-uniform hypergraph H, returns true if H is 3-valent.
def is_3_valent(H = IncidenceStructure([[]])):
return (Set([H.degree(i) % 3 for i in range(len(H.ground_set()))]) == Set([0]))
#Returns a list of all connected 3-uniform Veblen hypergraphs with exactly e edges, up to isomorphism.
def Veblen_3graphs(e=1):
V = []
for n in range(3, e+2): #might be able to give a better bound
for H in hypergraphs.nauty(e, n, uniform =3, multiple_sets = True, vertex_min_degree = 3, set_min_size = 3, connected = True):
if is_3_valent(IncidenceStructure(H)):
V.append(IncidenceStructure(H))
return V
len(Veblen_3graphs(e))
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