We prove that for every partially ordered set P , there exists c(P ) such that every family F of subsets of [n] ordered by inclusion and which contains no induced copy of P satisfies F ∈F 1/ n |F | ≤ c(P ). This confirms a conjecture of Lu and Milans [9].
Let
R
(
C
l
r
,
K
n
r
) be the Ramsey number of an
r‐uniform loose cycle of length
l versus an
r‐uniform clique of order
n. Kostochka et al. showed that for each fixed
r
≥
3, the order of magnitude of
R
(
C
3
r
,
K
n
r
) is
n
3
∕
2 up to a polylogarithmic factor in
n. They conjectured that for each
r
≥
3 we have
R
(
C
5
r
,
K
n
r
)
=
O
(
n
5
∕
4
). We prove that
R
(
C
5
3
,
K
n
3
)
=
O
(
n
4
∕
3
), and more generally for every
l
≥
3 that
R
(
C
l
3
,
K
n
3
)
=
O
(
n
1
+
1
∕
⌊
(
l
+
1
)
∕
2
⌋
). We also prove that for every
l
≥
5 and
r
≥
4,
R
(
C
l
r
,
K
n
r
)
=
O
(
n
1
+
1
∕
⌊
l
∕
2
⌋
) if
l is odd, which improves upon the result of Collier‐Cartaino et al. who proved that for every
r
≥
3 and
l
≥
4 we have
R
(
C
l
r
,
K
n
r
)
=
O
(
n
1
+
1
∕
(
⌊
l
∕
2
⌋
−
1
)
).
Let l (G) denote the list chromatic number of the r-uniform hypergraph G. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if G is simple and -regular, then l (G) ≥ (1∕(r − 1) + o(1)) log r . To see how close this inequality is to best possible, we examine l (G) when G is a random r-partite hypergraph with n vertices in each class. The value when r = 2 was determined by Alon and Krivelevich; here we show that l (G) = (g(r, ) + o(1)) log r almost surely, where is the expected average degree of G and = log n . The function g(r, ) is defined in terms of "preference orders" and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on l (G) for r = 2 and r = 3, but, perhaps surprisingly, apparently not for r ≥ 4.
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