A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game

We prove a tight lower bound on the asymptotic performance ratio $\rho$ of the bounded space online $d$-hypercube bin packing problem, solving an open question raised in 2005. In the classic $d$-hypercube bin packing problem, we are given a sequence of $d$-dimensional hypercubes and we have an unlimited number of bins, each of which is a $d$-dimensional unit hypercube. The goal is to pack (orthogonally) the given hypercubes into the minimum possible number of bins, in such a way that no two hypercubes in the same bin overlap. The bounded space online $d$-hypercube bin packing problem is a variant of the $d$-hypercube bin packing problem, in which the hypercubes arrive online and each one must be packed in an open bin without the knowledge of the next hypercubes. Moreover, at each moment, only a constant number of open bins are allowed (whenever a new bin is used, it is considered open, and it remains so until it is considered closed, in which case, it is not allowed to accept new hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448] showed that $\rho$ is $\Omega(\log d)$ and $O(d/\log d)$, and conjectured that it is $\Theta(\log d)$. We show that $\rho$ is in fact $\Theta(d/\log d)$. To obtain this result, we elaborate on some ideas presented by those authors, and go one step further showing how to obtain better (offline) packings of certain special instances for which one knows how many bins any bounded space algorithm has to use. Our main contribution establishes the existence of such packings, for large enough $d$, using probabilistic arguments. Such packings also lead to lower bounds for the prices of anarchy of the selfish $d$-hypercube bin packing game. We present a lower bound of $\Omega(d/\log d)$ for the pure price of anarchy of this game, and we also give a lower bound of $\Omega(\log d)$ for its strong price of anarchy

The Size-Ramsey Number of 3-uniform Tight Paths

Given a hypergraph H, the size-Ramsey number ˆr2(H) is the smallest integer m such that there exists a hypergraph G with m edges with the property that in any colouring of the edges of G with two colours there is a monochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices Pn(3) is linear in n, i.e., ˆr2(Pn(3)) = O(n). This answers a question by Dudek, La Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417–434], who proved ˆr2(Pn(3)) = O(n3/2 log3/2 n)

Partitioning by Monochromatic Trees

AbstractAnyr-edge-colouredn-vertex complete graphKncontains at mostrmonochromatic trees, all of different colours, whose vertex sets partition the vertex set ofKn, providedn⩾3r4r! (1−1/r)3(1−r)logr. This comes close to proving, for largen, a conjecture of Erdős, Gyárfás, and Pyber, which states thatr−1 trees suffice for alln

Near-optimum universal graphs for graphs with bounded degrees (Extended abstract)

Let H be a family of graphs. We say that G is H-universal if, for each H ∈H, the graph G contains a subgraph isomorphic to H. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an H(k, n)-universal graph Γ(k, n) with O(n2−2/k(log n)1+8/k) edges. This is optimal up to a small polylogarithmic factor, as Ω(n2−2/k) is a lower bound for the number of edges in any such graph. En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Γ(k, n), we prove, using a probabilistic argument, that Γ(k, n) is H(k, n)-universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties. © Springer-Verlag Berlin Heidelberg 200

The size-Ramsey number of powers of paths

Given graphs $G$ and $H$ and a positive integer $q$, say that $G$ \emph{is $q$-Ramsey for} $H$, denoted $G\rightarrow (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The \emph{size-Ramsey number} \sr(H) of a graph $H$ is defined to be \sr(H)=\min\{|E(G)|\colon G\rightarrow (H)_2\}. Answering a question of Conlon, we prove that, for every fixed~$k$, we have \sr(P_n^k)=O(n), where~$P_n^k$ is the $k$th power of the $n$-vertex path $P_n$ (i.e., the graph with vertex set $V(P_n)$ and all edges $\{u,v\}$ such that the distance between $u$ and $v$ in $P_n$ is at most $k$). Our proof is probabilistic, but can also be made constructive.Most of the work for this paper was done during my PhD, which was half funded by EPSRC grant reference 1360036, and half by Merton College Oxford. The third author was partially supported by FAPESP (Proc.~2013/03447-6) and by CNPq (Proc.~459335/2014-6, 310974/2013-5). The fifth author was supported by FAPESP (Proc.~2013/11431-2, Proc.~2013/03447-6 and Proc.~2018/04876-1) and partially by CNPq (Proc.~459335/2014-6). This research was supported in part by CAPES (Finance Code 001). The collaboration of part of the authors was supported by a CAPES/DAAD PROBRAL grant (Proc.~430/15)

Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox, and Zhao. These inheritance lemmas also imply improved H-counting lemmas for subgraphs of bijumbled graphs, for some H

Quasirandom permutations are characterized by 4-point densities

For permutations π and τ of lengths |π|≤|τ| , let t(π,τ) be the probability that the restriction of τ to a random |π| -point set is (order) isomorphic to π . We show that every sequence {τj} of permutations such that |τj|→∞ and t(π,τj)→1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→1/|π|! for every π ). This answers a question posed by Graham

On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi and Trotter states that there exists a constant c(\Delta) such that r(H) \leq c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The important open question is to determine the constant c(\Delta). The best results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta \log^2 \Delta}. We improve this upper bound, showing that there is a constant c for which c(\Delta) \leq 2^{c \Delta \log \Delta}. The induced Ramsey number r_{ind}(H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd\H{o}s conjectured the existence of a constant c such that, for any graph H on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in the exponent.Comment: 18 page

Arithmetic progressions in sets of fractional dimension

Let E\subset\rr be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$ contains non-trivial 3-term arithmetic progressions.Comment: 42 page