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We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.
Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.
We prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.
We answer four questions from a recent paper of Rao and Shinkar [17] on Lipschitz bijections between functions from {0, 1}n to {0, 1}. (1) We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch
$O(\sqrt{n})$
, matching a previously known lower bound. (3) We give a 3-Lipschitz embedding
$\phi \colon \{0,1\}^n \to \{0,1\}^{2n+1}$
such that
$${\rm{XOR }}(x) = {\rm{ Majority }}(\phi (x))$$
for all
$x \in \{0,1\}^n$
. (4) We show that with high probability there is an O(1)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.
Let I be a zero-dimensional ideal in the polynomial ring
$K[x_1,\ldots ,x_n]$
over a field K. We give a bound for the number of roots of I in
$K^n$
counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.
A k-permutation family on n vertices is a set-system consisting of the intervals of k permutations of the integers 1 to n. The discrepancy of a set-system is the minimum over all red–blue vertex colourings of the maximum difference between the number of red and blue vertices in any set in the system. In 2011, Newman and Nikolov disproved a conjecture of Beck that the discrepancy of any 3-permutation family is at most a constant independent of n. Here we give a simpler proof that Newman and Nikolov’s sequence of 3-permutation families has discrepancy
$\Omega (\log \,n)$
. We also exhibit a sequence of 6-permutation families with root-mean-squared discrepancy
$\Omega (\sqrt {\log \,n} )$
; that is, in any red–blue vertex colouring, the square root of the expected squared difference between the number of red and blue vertices in an interval of the system is
$\Omega (\sqrt {\log \,n} )$
.
Let
$\gamma(G)$
and
$${\gamma _ \circ }(G)$$
denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then
$$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$
In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then
$$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$
for some constant c independent of d. This result is sharp in the sense that as
$d \rightarrow \infty$
, almost all d-regular n-vertex graphs G of girth at least five have
Furthermore, if G is a disjoint union of
${n}/{(2d)}$
complete bipartite graphs
$K_{d,d}$
, then
${\gamma_\circ}(G) = \frac{n}{2}$
. We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that
${\gamma_\circ}(G) \sim {n}/{2}$
as
$d \rightarrow \infty$
. Therefore both the girth and regularity conditions are required for the main result.
A k-uniform tight cycle
$C_s^k$
is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if
$s \ge 2{k^2}$
and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of
$C_s^k$
. The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest.
For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For
$k \ge 3$
, there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then
$C_s^k$
is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for
$s \ge 5{k^2}$
, every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect
$C_s^k$
-tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.
We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs.
Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n – 2) have a tight Hamilton cycle.
Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d, α > 0 satisfying d + α > 1, any sufficiently large n-vertex such 3-graph H of density d and minimum 1-degree at least
$\alpha \left({\matrix{{n - 1} \cr 2 \cr } } \right)$
has a tight Hamilton cycle.
Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n.
Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition.
If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ.
If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m.
Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.
We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree
$\delta_1(G)>d$
then every vertex of G is contained in a copy of F in G?
We asymptotically determine c1(n, F) when F is the generalized triangle
$K_4^{(3)-}$
, and we give close to optimal bounds in the case where F is the tetrahedron
$K_4^{(3)}$
(the complete 3-graph on 4 vertices).
This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with
$m> n^2/4$
edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.
In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For
$r\geq 3$
we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.
Erdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if
$p = p(n) = \Omega(n^{-1/(2r)})$
, then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.
An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braı–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.
Given a fixed graph H, a real number p (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2n in G(n, 1/2) in nδ queries.
The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.
A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.
We show that, for a constant-degree algebraic curve γ in ℝD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of Charalambides [2]. This improves the earlier bound Ω(n5/4) of Charalambides [2].
We also show that, for every set P of n points that lie on a d-dimensional constant-degree algebraic variety V in ℝD, there exists a subset S ⊂ P of size at least Ω(n4/(9+12(d−1))), such that S spans
$\left({\begin{array}{*{20}{c}} {|S|} \\ 2 \\\end{array}} \right)$
distinct distances. This improves the earlier bound of Ω(n1/(3d)) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4].
Both results are consequences of a common technical tool.
For fixed graphs F1,…,Fr, we prove an upper bound on the threshold function for the property that G(n, p) → (F1,…,Fr). This establishes the 1-statement of a conjecture of Kohayakawa and Kreuter.
A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).
In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.