Counting substructures II: triple systems

For various triple systems $F$, we give tight lower bounds on the number of copies of $F$ in a triple system with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of Bollob\'as, Frankl, F\"uredi, Keevash, Pikhurko, Simonovits, and Sudakov who proved that there is one copy of $F$. A sample result is the following: F\"uredi-Simonovits and independently Keevash-Sudakov settled an old conjecture of S\'os by proving that the maximum number of triples in an $n$ vertex triple system (for $n$ sufficiently large and even) that contains no copy of the Fano plane is $p(n)={n/2 \choose 2}n.$ We prove that there is an absolute constant $c$ such that if $n$ is sufficiently large and $1 \le q \le cn^2$, then every $n$ vertex triple system with $p(n)+q$ edges contains at least $6q({n/2 \choose 4}+(n/2 -3){n/2 \choose 3}$$copies of the Fano plane. This is sharp for$q\le n/2-2$. Our proofs use the recently proved hypergraph removal lemma and stability results for the corresponding Tur\'an problem

Improved bounds for the Ramsey number of tight cycles versus cliques

The 3-uniform tight cycle $C_s^3$ has vertex set $Z_s$ and edge set $\{\{i, i+1, i+2\}: i \in Z_s\}$. We prove that for every $s \not\equiv 0$ (mod 3) and $s \ge 16$ or $s \in \{8,11,14\}$ there is a $c_s>0$ such that the 3-uniform hypergraph Ramsey number $$r(C_s^3, K_n^3)< 2^{c_s n \log n}$$ This answers in strong form a question of the author and R\"odl who asked for an upper bound of the form $2^{n^{1+\epsilon_s}}$ for each fixed $s \ge 4$, where $\epsilon_s \rightarrow 0$ as $s \rightarrow \infty$ and $n$ is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in $n$

Maximum H\mathcal H-free subgraphs

Given a family of hypergraphs $\mathcal H$, let $f(m,\mathcal H)$ denote the largest size of an $\mathcal H$-free subgraph that one is guaranteed to find in every hypergraph with $m$ edges. This function was first introduced by Erd\H{o}s and Koml\'{o}s in 1969 in the context of union-free families, and various other special cases have been extensively studied since then. In an attempt to develop a general theory for these questions, we consider the following basic issue: which sequences of hypergraph families $\{\mathcal H_m\}$ have bounded $f(m,\mathcal H_m)$ as $m\to\infty$? A variety of bounds for $f(m,\mathcal H_m)$ are obtained which answer this question in some cases. Obtaining a complete description of sequences $\{\mathcal H_m\}$ for which $f(m,\mathcal H_m)$ is bounded seems hopeless

Sparse hypergraphs with low independence number

Let K_4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erd\H{o}s, Koml\'os, and Szemer\'edi (1981) asked if there is a function \omega(d) tending to infinity such that every 3-uniform, K_4-free hypergraph N vertices and average degree d has independence number at least \omega(d) N/d^{1/2}. We answer this question by constructing a 3-uniform, K_4-free hypergraph with independence number at most 2N/d^{1/2}. We also provide counterexamples to several related conjectures and improve the lower bound of some hypergraph Ramsey numbers

A survey of hypergraph Ramsey problems

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments and open problems

Biclique Coverings and the Chromatic Number

Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges of $G$, then their number is at least $2^{\sqrt{\log_2 k}}$. This is the first improvement of the easy lower bound of $\log_2 k$, while the Alon-Saks-Seymour conjecture states that this can be improved to $k-1$. \medskip \noindent $\bullet$ The sum of the orders of the bicliques is at least $(1-o(1))k\log_2 k$. This generalizes, in asymptotic form, a result of Katona and Szemer\'edi who proved that the minimum is $k\log_2 k$ when $G$ is a clique

Hypergraph Ramsey numbers: tight cycles versus cliques

For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding to $s$ distinct points on a circle and edge set given by the $s$ cyclic intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$ (mod 3) we prove that there are positive constants $a$ and $b$ with $$2^{at}<r(C^3_s, K^3_t)<2^{bt^2\log t}.$$ The lower bound is obtained via a probabilistic construction. The upper bound for $s>5$ is proved by using supersaturation and the known upper bound for $r(K_4^{3}, K_t^3)$, while for $s=5$ it follows from a new upper bound for $r(K_5^{3-}, K_t^3)$ that we develop

An extremal graph problem with a transcendental solution

We prove that the number of multigraphs with vertex set $\{1, \ldots, n\}$ such that every four vertices span at most nine edges is $a^{n^2 + o(n^2)}$ where $a$ is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy-Schwarz arguments. Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers

Almost all cancellative triple systems are tripartite

A triple system is cancellative if no three of its distinct edges satisfy $A \cup B=A \cup C$. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is cancellative. We prove that almost all cancellative triple systems with vertex set [n] are tripartite. This sharpens a theorem of Nagle and Rodl on the number of cancellative triple systems. It also extends recent work of Person and Schacht who proved a similar result for triple systems without the Fano configuration. Our proof uses the hypergraph regularity lemma of Frankl and Rodl, and a stability theorem for cancellative triple systems due to Keevash and the second author

Polynomial to exponential transition in Ramsey theory

Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices span at most $t$ edges. Erd\H os and Hajnal conjectured (1972) that $h^{(k)}(s)$ can be calculated precisely using a recursive formula and Erd\H os offered \$500 for a proof of this. For$k=3$this has been settled for many values of$s$including powers of three but it was not known for any$k\geq 4$and$s\geq k+2$. Here we settle the conjecture for all$s \ge k \ge 4$. We also answer a question of Bhat and R\"odl by constructing, for each$k \ge 4$, a quasirandom sequence of$k$-uniform hypergraphs with positive density and upper density at most$k!/(k^k-k)$. This result is sharp.Comment: 27 page