Fano plane
Let H = (V, E) be the complete 4-partite hypergraph with the vertex partition V =
V1∪V˙ 2∪V˙ 3∪V˙ 4 of almost equal size: ||Vi| − |Vj|| ≤ 1 for 1 ≤ i < j ≤ 4. We color its
hyperedges with colors from [r] as follows. The hyperedges from E(V1∪ V3, V2∪ V4) can
be colored with colors from {1, . . . , r − 2}, from E(V1 ∪ V2, V3 ∪ V4) with color r − 1
and from E(V1∪ V4, V2∪ V3) with color r. Obviously, there are no monochromatic Fano
planes, as all monochromatic induced subhypergraphs are bipartite. It remains to verify a lower bound on the number of possible colorings (we now assume for simplicity that 4 divides n):
• the hyperedges that intersect 3 of the possible 4 partition classes can be colored arbitrarily (i.e., by r colors), which gives
r4(n4) 3
colorings for those hyperedges,
• the hyperedges from E(V1, V2), E(V1, V4), E(V2, V3) or E(V3, V4) can be colored
with r − 1 colors and since e(Vi, Vj) = 2 n/42 n4 we obtain:
(r − 1)4·2(n/42 )
n
4
5.3 Using more than 3 colors colorings for these hyperedges,
• the hyperedges from E(V1, V3) or E(V2, V4) can be colored with 2 colors in
22·2(n/42 ) n 4 many ways. Consequently, c4,F(n) ≥ r4( n 4) 3 (r − 1)4·2(n/42 ) n 422·2( n/4 2 ) n 4 ≥ q√ 2r(r − 1) n3/8−O(n2) ≥ (r + ε)e(Bn)
for any r ≥ 4 and for some ε > 0 and sufficiently large n.
We note that this lower bound on the number of Fano plane-free r-colorings can be easily improved. For example, if one distributes the available colors for the three bipartitions as evenly as possible, then one obtains the following for r ≥ 4
cr,F(n) ≥ fn 3/8−O(n2) r , with fr= 2 3 3/4 r5/4 if r = 0 mod 3 r1/2l23rm1/2j23rk1/4 if r = 1 mod 3 r1/2l23rm1/4j23rk1/2 if r = 2 mod 3.
Generalized triangles T3 and T4
Below we prove lower bounds cr,T3(n) r
ex(n,T3) and c
r,T4(n) r
ex(n,T4) for r ≥ 4. In
the following we assume for simplicity that n is divisible by 3 and 4.
First we consider the case of the 3-uniform generalized triangle T3. To prove a lower bound on cr,T3(n) we give a lower bound on cr,K3(n), i.e., for the case of graphs, where
we forbid a monochromatic triangle, see also [ABKS04]. Namely, consider the following graph G = (V, E) with |V | = n vertices. Let V = V1∪V˙ 2∪V˙ 3∪V˙ 4 be a partition of the vertex set V with |Vi| = n/4, i ∈ [4]. The edge set E of G consists of all edges e = {v, w}
with v ∈ Vi and w ∈ Vj, where i 6= j. Given the set [r], r ≥ 4, of colors, we color the set
of all edges between classes V1 and V2, or between V3 and V4 by the colors 1, . . . , r − 1. For the set of all edges between the classes V1 and V4, or V2 and V3 we use the colors
1, . . . , r − 2, r. Moreover, the set of all edges between the classes V1 and V3, or V2 and
V4 are colored arbitrarily by the colors r − 1 and r. Here every coloring gives rise to a
monochromatic bipartite graph, so no monochromatic triangle is created by the colorings described above.
The number of these colorings in G for r ≥ 4 is
cr,K3(n) ≥ cr,K3(G) = (r − 1) 4(n 4) 2 · 22(n4) 2 =(r − 1) · √ 2 n2 4 rn24 ≥ rex(n,K3). (5.103) The lower bound (5.103) may be improved by using another distribution of the set [r] of colors, namely for r divisible by 3 say, we color the set of all edges between the classes
5 Restricted edge colorings of hypergraphs
V1 and V2, or V3 and V4 by the colors 1, . . . , 2r/3. For the set of all edges between the
classes V1 and V4, or V2 and V3 we use the colors 1, . . . , r/3, 2r/3 + 1, . . . , r. Moreover,
the set of all edges between the classes V1 and V3, or V2 and V4 are colored arbitrarily by the colors r/3 + 1, . . . , r, which gives
cr,K3(n) ≥ 2r 3 3 2 !n24 rex(n,K3) (5.104) colorings.
Now we consider the 3-uniform generalized triangle T3 and the 3-uniform, 2-partite hypergraph H3 = (V, E) on |V | = n vertices, which is defined as follows. Let V = V0∪V˙ 0
be a partition with |V0| = n/3 and |V0| = 2n/3. All hyperedges e ∈ E contain exactly
one vertex from V0 and two vertices from V0. On the set V0 we place the graph G from above with m = 2n/3 vertices. For any hyperedge e = {v0, v, w} ∈ E with e ∩ V0 = {v0}
its link {v, w} has to be an edge in the graph G. The hyperedge e = {v0, v, w} may be
colored by some color by which the edge {v, w} may be colored. Using (5.103), this yields
cr,T3(n) ≥ cr,T3(H3) = (r − 1) ·√2 (2n/3)2 4 ! n 3 =(r − 1) ·√2 n3 27 rn327 ≥ rex(n,T3) (5.105)
colorings for r ≥ 4 and n sufficiently large. Of course, (5.105) may be improved by using (5.104).
It remains to show that the hypergraph H3 does not contain a generalized triangle
T3. If {a, b, c}, {b, c, d} and {a, d, e} is a subhypergraph T3 in H3, then one of the two
vertices b or c, and e must be contained in class V0, say b, e ∈ V0. But then the union
of the links of the vertices b and d forms a triangle in the graph G. However, due to the construction of the colorings, there is no monochromatic triangle T2 in G, hence no monochromatic triangle T3.
Next we consider the 4-uniform generalized triangle T4 and the 4-uniform, 2-partite hypergraph H4 = (V, E) on |V | = n vertices, which is defined as follows. Let V = V0∪V˙ 0
be a partition with |V0| = n/4 and |V0| = 3n/4. All hyperedges e ∈ E contain exactly
one vertex from V0 and three vertices from V0. On the set V0 we place the hypergraph
H3 from above with m = 3n/4 vertices. For any hyperedge e = {v0, v, w, x} ∈ E
with e ∩ V0 = {v0} its link {v, w, x} has to be a hyperedge in the hypergraph H3.
The hyperedge e = {v0, v, w, x} may be colored by some color by which the hyperedge
{v, w, x} in H3 may be colored.
5.3 Using more than 3 colors With (5.105), this gives
cr,T4(n) ≥ cr,T4(H4) = (r − 1) ·√2 (3n/4)3 27 ! n 4 =(r − 1) ·√2 n4 256 r256n4 ≥ rex(n,T4)
colorings for r ≥ 4 and n sufficiently large.
It remains to show that the hypergraph H4 does not contain a generalized triangle
T4. If {a, b, c, d}, {e, b, c, d} and {a, e, f, g} is a subhypergraph T4 in H4, then one of the
three vertices b, c or d, and f or g must be contained in class V0, say b, f ∈ V0. But then the union of the links of b and f forms a generalized triangle in the hypergraph H3. However, due to the construction of the colorings, there is no monochromatic generalized triangle T3, hence no monochromatic generalized triangle T4.
Expanded complete graph and Fan(k)-hypergraph
We show for fixed r ≥ 4 for the expanded complete graph H`+1k and for the Fan(k) hypergraph F`+1k the lower bound which is exponentially larger than re(T`(k)(n)) for n
sufficiently large.
Let V be an n-element vertex set and we assume for simplicity that 2` divides n. Consider a partition P of the vertex set V into (` + 2) pairwise disjoint vertex sets
V1, . . . , V`−2, W1, . . . , W4, where each class Vi, i ∈ [` − 2], has cardinality |Vi| = n/`,
and every other class Wi, i ∈ [4], satisfies |Wi| = n/(2 · `). Let H be the k-uniform (` + 2)-partite hypergraph with respect to the partition P, where all crossing hyperedges are present except for those that intersect more than two classes Wi, Wj, i 6= j. Let
{1, . . . , r} be the set of colors.
All hyperedges in E(H) which contain at most one vertex from W1 ∪ · · · ∪ W4 can be colored with all r colors. All hyperedges in E(H) which contain one vertex from each class W1 and W2 or from each class W3 and W4 are colored with 1, . . . , r − 1. All hyperedges in E(H) which contain one vertex from each class W1 and W3 or from each class W2 and W4 get colors 1, . . . , r − 2, r. All hyperedges in E(H) which contain one
vertex from each class W1and W4or from each class W2and W3 are colored with r −1, r. Note that the projection (link) of any three hyperedges on W1∪ · · · ∪ W4 does not give a monochromatic graph triangle.
Then, the number of colorings of the set E(H) of hyperedges of H is precisely
r(`−2k )(n/`)k+2( `−2 k−1)(n/`)k · (r − 1)( `−2 k−2)(n/`)k· 2(1/2)( `−2 k−2)(n/`)k = r(k`) · (r − 1) ·√2( `−2 k−2) r( `−2 k−2) (n/`)k r(k`)(n/`) k ≥ rex(n,H`+1k )
5 Restricted edge colorings of hypergraphs
Suppose for contradiction that for one of these colorings the hypergraph H contains a monochromatic H`+1k with core v1, . . . , v`+1. As all hyperedges in H are crossing, at least
three vertices of the core of H`+1k must be contained in the set W1∪ · · · ∪ W4. Without loss of generality let v1, v2, v3 be such vertices. By construction, no two of these can be
contained in the same vertex set Wi. For each pair {vi, vj}, 1 ≤ i < j ≤ 3, there is a
(k − 2)-element set Si,j such that {vi, vj} ∪ Si,j is a hyperedge in H`+1k , and again by
construction Si,j ⊆ V1 ∪ · · · ∪ V`−2, but then the links LH(Si,j), 1 ≤ i < j ≤ 3, yield a graph triangle in W1∪ · · · ∪ W4, hence the hyperedges {vi, vj} ∪ Si,j, 1 ≤ i < j ≤ 3, do
not have all the same color.
Now assume that we obtain a monochromatic subhypergraph F`+1k with core vertices
v1, . . . , v`+1 where v1, . . . , vk form a hyperedge in F`+1k . Then at least three of the core
vertices must be contained in the set W1∪ · · · ∪ W4, say these are the vertices vg, vh, vi,
where 1 ≤ g < h < i. We must have g ≤ k as otherwise we proceed similarly to the paragraph above to obtain a contradiction. Moreover, by construction we cannot have
i ≤ k, as v1, . . . , vkform a hyperedge of F`+1k . If g, h ≤ k, then with the sets Sg,i and Sh,i
forming a hyperege with {g, i} and {h, i}, respectively, the links L({v1, . . . , vk}\{vg, vh}),
L(Sg,i), and L(Sh,i) yield a monochromatic triangle in W1∪· · ·∪W4, which is not possible.
On the other hand, if h ≥ k + 1, the same reasoning applies, and we are finished. The lower bound can be improved for larger values of r by better distributing the colors, similarly to (5.104), which gives (for r divisible by 3) the lower bound
cr,Hk `+1 (n), cr,Fk `+1 (n) ≥ r( ` k) · 2 · √ 2r 3√3 !(k−2`−2) (n/`)k . (5.106)