The Ramsey Number for 3-Uniform Tight Hypergraph Cycles Article

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles 191
vertices in Vblue = V contains a copy of C (3)
4 in C1 red or C2 red . Hence, we can find greedily,
by taking one edge from a copy of C (3)
4 and re-using the remaining vertex, a matching of
size
� √ � � √ √ �
t − 25 t /3 � 6s + c0 s − 25 7s /3 � 2s
in C1 red ∪ C2 red . Thus, there is an index i ∈{1, 2} such that Ci red contains M(3) s as well as a
copy of C (3)
4 .
Assume now that Vred �= ∅ and, thus, S exists and C1 red = S. We know (see Observation
7.5) that A contains a matching MA, V (MA) ⊂ Vblue, of size s but no C (3)
4 .Asin
the proof of Observation 7.2, for every vertex x ∈ Vred and each edge e ∈ MA, there
exists a edge f ∈ S so that x ∈ f and |e ∩ f| = 2. Hence, we can find a matching
of size |Vred|