Storage architectures ranging from minimum bandwidth regenerating encoded distributed storage systems to declustered-parity RAIDs can employ dense partial Steiner systems to support fast reads, writes, and recovery of failed storage units. To enhance performance, popularities of the data items should be taken into account to make frequencies of accesses to storage units as uniform as possible. A combinatorial model ranks items by popularity and assigns data items to elements in a dense partial Steiner system so that the sums of ranks of the elements in each block are as equal as possible. By developing necessary conditions in terms of independent sets, we demonstrate that certain Steiner systems must have a much larger difference between the largest and smallest block sums than is dictated by an elementary lower bound. In contrast, we also show that certain dense partial 
$$S(t,t+1,v)$$

 designs can be labeled to realize the elementary lower bound. Furthermore, we prove that for every admissible order v, there is a Steiner triple system (S(2, 3, v)) whose largest difference in block sums is within an additive constant of the lower bound.

Access balancing in storage systems by labeling partial Steiner systems

Storage architectures ranging from minimum bandwidth regenerating encoded distributed storage systems to declustered-parity RAIDs can be designed using dense partial Steiner systems in order to support fast reads, writes, and recovery of failed storage units. In order to ensure good performance, popularities of the data items should be taken into account and the frequencies of accesses to the storage units made as uniform as possible. A proposed combinatorial model ranks items by popularity and assigns data items to elements in a dense partial Steiner system so that the sums of ranks of the elements in each block are as equal as possible. By developing necessary conditions in terms of independent sets, we demonstrate that certain Steiner systems must have a much larger difference between the largest and smallest block sums than is dictated by an elementary lower bound. In contrast, we also show that certain dense partial $S(t, t+1, v)$ designs can be labeled to realize the elementary lower bound. Furthermore, we prove that for every admissible order $v$, there is a Steiner triple system $(S(2, 3, v))$ whose largest difference in block sums is within an additive constant of the lower bound.

Access Balancing in Storage Systems by Labeling Partial Steiner Systems

Storage architectures ranging from minimum bandwidth regenerating encoded distributed storage systems to declustered-parity RAIDs can employ dense partial Steiner systems to support fast reads, writes, and recovery of failed storage units. To enhance performance, popularities of the data items should be taken into account to make frequencies of accesses to storage units as uniform as possible. A combinatorial model ranks items by popularity and assigns data items to elements in a dense partial Steiner system so that the sums of ranks of the elements in each block are as equal as possible. By developing necessary conditions in terms of independent sets, we demonstrate that certain Steiner systems must have a much larger difference between the largest and smallest block sums than is dictated by an elementary lower bound. In contrast, we also show that certain dense partial S(t, t+1, v) designs can be labeled to realize the elementary lower bound. Furthermore, we prove that for every admissible order v, there is a Steiner triple system (S (2, 3, v)) whose largest difference in block sums is within an additive constant of the lower bound.

/pdf/access-balancing-in-storage-systems-by-labeling-partial-40ue22nw0e.pdf

A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n (\log t)^{1/(k-1)}/t$ for some $c>0$. In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3 and 4. Our work is motivated by a problem of Lo and Zhao, who asked for $k\ge 4$, how large of an independent set a $k$-graph $H$ on $n$ vertices necessarily has when its maximum $(k-2)$-degree $\Delta_{k-2}(H)\le dn$. (The corresponding problem with respect to $(k-1)$-degrees was solved by Kostochka, Mubayi, and Varstra\"ete [Random Structures&Algorithms 44, 224--239, 2014].) In this paper we show that every $k$-graph $H$ on $n$ vertices with $\Delta_{k-2}(H)\le dn$ contains an independent set of size $c (\frac nd \log\log \frac nd)^{1/(k-1)}$, and under additional conditions, an independent set of size $c (\frac nd \log \frac nd)^{1/(k-1)}$. The former assertion gives a new upper bound for the $(k-2)$-degree Tur\'an density of complete $k$-graphs.

/pdf/independence-number-of-hypergraphs-under-degree-conditions-3hz1ficz.pdf

Independence number of hypergraphs under degree conditions

A k-uniform hypergraph with n vertices is an (n,k,l)-omitting system if it has no two edges with intersection size l. If in addition it has no two edges with intersection size greater than l, then it is an (n,k,l)-system. Rodl and Sinajova proved a sharp lower bound for the independence number of (n,k,l)-systems. We consider the same question for (n,k,l)-omitting systems. Our proofs use adaptations of the random greedy independent set algorithm, and pseudorandom graphs. We also prove related results where we forbid more than two edges with a prescribed common intersection size leading to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number r(k)(F,t), where F is the k-uniform Fan. The behavior is quite different than the case k=2 which is the classical Ramsey number r(3,t).YY

Independent Sets in Hypergraphs Omitting an Intersection

Given positive integers k, $$\ell $$
 and $$\lambda $$
 such that $$2\leqslant \ell \leqslant k-1$$
 , an $$(n,k,\ell ,\lambda )$$
 -hypergraph $$\mathcal {H}$$
 is a k-uniform hypergraph on the vertex set [n] in which every subset of size $$\ell $$
 is contained in at most $$\lambda $$
 edges. The independence number $$\alpha (\mathcal {H})$$
 of $$\mathcal {H}$$
 is the maximum size of a subset of vertices which contains no edges of $$\mathcal {H}$$
 . Let $$f(n,k,\ell ,\lambda )=\min \{\alpha (\mathcal {H})\mid \mathcal {H}\ \mathrm{{is\ an\ }}(n,k,\ell ,\lambda )$$
 -hypergraph
 $$\}$$
 . In this paper we show that for any given positive integers $$k\geqslant 5$$
 , $$\frac{2k+4}{5}<\ell \leqslant k-2$$
 and $$\lambda \leqslant \left\lfloor \dfrac{n^{\frac{5\ell -2k-4}{3k-9}}}{\omega (n)}\right\rfloor $$
 , $$\begin{aligned} f(n,k,\ell ,\lambda )\geqslant C_{k,\ell }\left( \frac{n}{\lambda }\log \frac{n}{\lambda }\right) ^{\frac{1}{\ell }}, \end{aligned}$$
 where $$\omega (n)\rightarrow \infty $$
 arbitrarily slowly as $$n\rightarrow \infty $$
 and $$C_{k,\ell }$$
 is a constant depending only on k and $$\ell $$
 . In particular, $$C_{k,\ell }\sim \frac{\ell -1}{e}$$
 as $$k\rightarrow \infty $$
 . It generalizes the results from the case $$\ell =k-1$$
 or $$\lambda =1$$
 . An upper bound on $$f(n,k,\ell ,\lambda )$$
 is also obtained for $$k\geqslant 3$$
 , $$2\leqslant \ell \leqslant k-1$$
 and $$\log n\ll \lambda \ll n$$
 , by considering a random k-uniform hypergraph $$\mathcal {H}_k(n,p)$$
 .

Bounding the Independence Number in Some $$(n,k,\ell ,\lambda )$$ ( n , k , ℓ , λ ) -Hypergraphs

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Bounding the Independence Number in Some $$(n,k,\ell ,\lambda )$$ ( n , k , ℓ , λ ) -Hypergraphs