Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Mubayi's Conjecture and conditionally intersecting sets   Mubayi's Conjecture states that if $\mathcal{F}$ is a family of $k$-sized
subsets of $[n] = \{1,\ldots,n\}$ which, for $k \geq d \geq 2$, satisfies $A_1
\cap\cdots\cap A_d \neq \emptyset$ whenever $|A_1 \cup\cdots\cup A_d| \leq 2k$
for all distinct sets $A_1,\ldots,A_d \in\mathcal{F}$, then $|\mathcal{F}|\leq
\binom{n-1}{k-1}$, with equality occurring only if $\mathcal{F}$ is the family
of all $k$-sized subsets containing some fixed element. This paper proves that
Mubayi's Conjecture is true for all families that are invariant with respect to
shifting; indeed, these families satisfy a stronger version of Mubayi's
Conjecture. Relevant to the conjecture, we prove a fundamental bijective
duality between $(i,j)$-unstable families and $(j,i)$-unstable families.
Generalising previous intersecting conditions, we introduce the
$(d,s,t)$-conditionally intersecting condition for families of sets and prove
general results thereon. We conjecture on the size and extremal structures of
families $\mathcal{F}\in\binom{[n]}{k}$ that are $(d,2k)$-conditionally
intersecting but which are not intersecting, and prove results related to this
conjecture. We prove fundamental theorems on two $(d,s)$-conditionally
intersecting families that generalise previous intersecting families, and we
pose an extension of a previous conjecture by Frankl and F\"uredi on
$(3,2k-1)$-conditionally intersecting families. Finally, we generalise a
classical result by Erd\H{o}s, Ko and Rado by proving tight upper bounds on the
size of $(2,s)$-conditionally intersecting families $\mathcal{F}\subseteq
2^{[n]}$ and by characterising the families that attain these bounds. We extend
this theorem for certain parametres as well as for sufficiently large families
with respect to $(2,s)$-conditionally intersecting families
$\mathcal{F}\subseteq 2^{[n]}$ whose members have at most a fixed number $u$
members.

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The categories of this paper are: comp-gas
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: eess