Abstract
In this paper we study generalizations of classical results on intersection patterns of set systems in
R
d
$\mathbb {R}^d$
double struck upper R Superscript d
, such as the fractional Helly theorem or the
(
p
,
q
)
$(p,q)$
left parenthesis p comma q right parenthesis
-theorem, in the setting of arbitrary triangulable spaces with a forbidden homological minor.
Given a simplicial complex
K
and an integer
b
, we say that a family
F
$\mathcal {F}$
script upper F
of subcomplexes of some simplicial complex
X
is a
(K,b)-free cover
if (i)
K
is a forbidden homological minor of
X
, and (ii) the
j
th reduced Betti number
β
~
j
(
⋂
S
∈
G
S
,
Z
2
)
$\tilde {\beta }_j(\bigcap _{S\in {\mathcal {G}}}S,\mathbb {Z}_2)$
ModifyingAbove beta With tilde Subscript j Baseline left parenthesis intersection Underscript upper S element of script upper G Endscripts upper S comma double struck upper Z 2 right parenthesis
is strictly less than
b
for all
0
≤
j
<
dim
K
$0\leq j < \dim K$
0 less than or equals j less than dimension upper K
and all nonempty subfamilies
G
⊆
F
$\mathcal {G}\subseteq \mathcal {F}$
script upper G subset of or equal to script upper F
.
We show that for every
K
and
b
, the fractional Helly number of a
(
K
,
b
)
$(K,b)$
left parenthesis upper K comma b right parenthesis
-free cover is at most
μ
(
K
)
+
1
$\mu (K)+1$
mu left parenthesis upper K right parenthesis plus 1
, where
μ
(
K
)
$\mu (K)$
mu left parenthesis upper K right parenthesis
is the maximum sum of the dimensions of two disjoint faces in
K
. This implies that the assertion of the
(
p
,
q
)
$(p,q)$
left parenthesis p comma q right parenthesis
-theorem holds for every
p
≥
q
>
μ
(
K
)
$p \ge q> \mu (K)$
p greater than or equals q greater than mu left parenthesis upper K right parenthesis
and every
(
K
,
b
)
$(K,b)$
left parenthesis upper K comma b right parenthesis
-free cover
F
$\mathcal {F}$
script upper F
. For
b
=
1
$b=1$
b equals 1
and a suitable
K
, this recovers the original
(
p
,
q
)
$(p,q)$
left parenthesis p comma q right parenthesis
-theorem and its generalization to good covers. Interestingly, our results show that the range of parameters
(
p
,
q
)
$(p,q)$
left parenthesis p comma q right parenthesis
for which the
(
p
,
q
)
$(p,q)$
left parenthesis p comma q right parenthesis
-theorem holds is independent of
b
.
Our proofs use Ramsey-type arguments combined with the notion of stair convexity of Bukh et al. to construct (forbidden) homological minors in certain cubical complexes.