Hypergraph Theorems
Adithya M S
April 20, 2025
Problem/Theorem #: In a k-uniform hypergraph, if there are less than 2k−1
edges it can be colored with just 2-colors, leaving no edge monochromatic
Proof. We color the vertices of the hypergraph randomly with one of the two
colors. Consider an edge e of the hypergraph. What is the probability that e is
monochromatic ?
It is given by,
P[edge eis monochromatic] = 2 ×1
2k
=1
2k−1
By the union bound, we have,
P[some edge eis monochromatic] ⩽X
e∈E(G)
P[edge eis monochromatic]
<2k−1.(1
2)k−1= 1
Since, probability that some edge e is monochromatic is less than 1, there is
a non-zero probability that no edge e is monochromatic and hence there exists
a coloring of the hypergraph G such that no edge e is monochromatic.
♢
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