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Register a new userOn the Number of Independent Sets in Uniform, Regular, Linear Hypergraphs
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We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up to the first order term for all (fixed) $rge 3$, with $d$ and $n$ going to infinity. In the case of strong independent sets, for $r=3$, we provide an upper bound that is tight up to the second-order term, improving on a result of Ordentlich-Roth (2004). The tightness in the strong independent set case is established by an explicit construction of a $3$-uniform, $d$-regular, cross-edge free, linear hypergraph on $n$ vertices which could be of interest in other contexts. We leave open the general case(s) with some conjectures. Our proofs use the occupancy method introduced by Davies, Jenssen, Perkins, and Roberts (2017).
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Let $kge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $binom{n-1}{k-1} + lfloorfrac{n-1}{k} rfloor$, and the equality holds if and only if $H$ is a full $k$-star with center $v$ together with a maximal matching omitting $v$. This verifies a conjecture of Mubayi and Verstra{e}te.
A $k$-uniform hypergraph with $n$ vertices is an $(n,k,ell)$-omitting system if it does not contain two edges whose intersection has size exactly $ell$. If in addition it does not contain two edges whose intersection has size greater than $ell$, then it is an $(n,k,ell)$-system. R{o}dl and v{S}iv{n}ajov{a} proved a lower bound for the independence number of $(n,k,ell)$-systems that is sharp in order of magnitude for fixed $2 le ell le k-1$. We consider the same question for the larger class of $(n,k,ell)$-omitting systems. For $kle 2ell+1$, we believe that the behavior is similar to the case of $(n,k,ell)$-systems and prove a nontrivial lower bound for the first open case $ell=k-2$. For $k>2ell+1$ we give new lower and upper bounds which show that the minimum independence number of $(n,k,ell)$-omitting systems has a very different behavior than for $(n,k,ell)$-systems. Our lower bound for $ell=k-2$ uses some adaptations of the random greedy independent set algorithm, and our upper bounds (constructions) for $k> 2ell+1$ are obtained from some pseudorandom graphs. We also prove some related results where we forbid more than two edges with a prescribed common intersection size and this leads to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number $r_{k}(F^{k},t)$, where $F^{k}$ is the $k$-uniform Fan. Here the behavior is quite different than the case $k=2$ which reduces to the classical graph Ramsey number $r(3,t)$.
Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for the so-called coregular k-uniform hypergraphs. In this paper, we extend this characterization to all k-uniform hypergraphs, not just the coregular ones. Specifically, we prove that if a k-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then there is a gap between its first and second largest eigenvalue.
An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.
We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing with data. Let H be such a hypergraph on a set of vertices V. We give a lower bound of 1.4977 |V | and an upper bound of 1.5012 |V | .
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