Let T be a tree rooted at r . Two vertices of T are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets A and B of V ( T ) are unrelated if, for any a ? A and b ? B , a and b are unrelated. Let ω be a nonnegative weight function defined on V ( T ) with ? v ? V ( T ) ω ( v ) = 1 . In this note, we prove that either there is an ( r , u ) -path P with ? v ? V ( P ) ω ( v ) ? 1 3 for some u ? V ( T ) , or there exist unrelated sets A , B ? V ( T ) such that ? a ? A ω ( a ) ? 1 3 and ? b ? B ω ( b ) ? 1 3 . The bound 1 3 is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomasse.
[1]
Marthe Bonamy,et al.
The Erd\H{o}s-Hajnal Conjecture for Long Holes and Anti-holes
,
2014
.
[2]
Paul Erdös,et al.
Ramsey-type theorems
,
1989,
Discret. Appl. Math..
[3]
Marthe Bonamy,et al.
The Erdös-Hajnal Conjecture for Long Holes and Antiholes
,
2014,
SIAM J. Discret. Math..
[4]
Nicolas Bousquet,et al.
The Erdős-Hajnal conjecture for paths and antipaths
,
2015,
J. Comb. Theory, Ser. B.
[5]
Maria Chudnovsky,et al.
The Erdös–Hajnal Conjecture—A Survey
,
2014,
J. Graph Theory.