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Given $r$-uniform hypergraphs $G$ and $H$ the Turan number $rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the ErdH{o}s-Renyi random $r$-uniform hypergraph, and $H=C_{2ell}^{(r)}$, the $r$-uniform linear cycle of length $2ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $rm ex(G_{n,p}^{(r)}, C_{2ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $ell$ grows. Our main technical contribution is a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the supersaturation result depends on the codegree of some pairs of vertices in the underlying hypergraph. This approach could be used to prove similar results for other hypergraphs $H$.
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Let $t$ be an integer such that $tgeq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples ${a,x_i,y_i}$, ${b,x_i,y_i}$ for $1 le i le t$, where the elements $a, b, x_1, x_2, ldots, x_t,$ $y_1, y_2, ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstraete, where the special case $t=2$ was a problem of ErdH{o}s that was studied by various authors. Mubayi and Verstraete proved that $ex(n,K_{2,t}^{(3)})<t^4binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})geq frac{2t-1}{3} binom{n}{2}$. These bounds together with a standard argument show that $g(t):=lim_{nto infty} ex(n,K_{2,t}^{(3)})/binom{n}{2}$ exists and that [frac{2t-1}{3}leq g(t)leq t^4.] Addressing the question of Mubayi and Verstraete on the growth rate of $g(t)$, we prove that as $t to infty$, [g(t) = Theta(t^{1+o(1)}).]
Let the bipartite Turan number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any positive integers $m,n,t$ with $ngeq mgeq tgeq frac{m}{2}+1$. This confirms the rest of a conjecture of Gy{o}ri cite{G97} (in a stronger form), and improves the upper bound of $ex(m,n,C_{2t})$ obtained by Jiang and Ma cite{JM18} for this range. We also prove a tight edge condition for consecutive even cycles in bipartite graphs, which settles a conjecture in cite{A09}. As a main tool, for a longest cycle $C$ in a bipartite graph, we obtain an estimate on the upper bound of the number of edges which are incident to at most one vertex in $C$. Our two results generalize or sharpen a classical theorem due to Jackson cite{J85} in different ways.
We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.
A {em special four-cycle } $F$ in a triple system consists of four triples {em inducing } a $C_4$. This means that $F$ has four special vertices $v_1,v_2,v_3,v_4$ and four triples in the form $w_iv_iv_{i+1}$ (indices are understood $pmod 4$) where the $w_j$s are not necessarily distinct but disjoint from ${v_1,v_2,v_3,v_4}$. There are seven non-isomorphic special four-cycles, their family is denoted by $cal{F}$. Our main result implies that the Turan number $text{ex}(n,{cal{F}})=Theta(n^{3/2})$. In fact, we prove more, $text{ex}(n,{F_1,F_2,F_3})=Theta(n^{3/2})$, where the $F_i$-s are specific members of $cal{F}$. This extends previous bounds for the Turan number of triple systems containing no Berge four cycles. We also study $text{ex}(n,{cal{A}})$ for all ${cal{A}}subseteq {cal{F}}$. For 16 choices of $cal{A}$ we show that $text{ex}(n,{cal{A}})=Theta(n^{3/2})$, for 92 choices of $cal{A}$ we find that $text{ex}(n,{cal{A}})=Theta(n^2)$ and the other 18 cases remain unsolved.
We study hypergraph discrepancy in two closely related random models of hypergraphs on $n$ vertices and $m$ hyperedges. The first model, $mathcal{H}_1$, is when every vertex is present in exactly $t$ randomly chosen hyperedges. The premise of this is closely tied to, and motivated by the Beck-Fiala conjecture. The second, perhaps more natural model, $mathcal{H}_2$, is when the entries of the $m times n$ incidence matrix is sampled in an i.i.d. fashion, each with probability $p$. We prove the following: 1. In $mathcal{H}_1$, when $log^{10}n ll t ll sqrt{n}$, and $m = n$, we show that the discrepancy of the hypergraph is almost surely at most $O(sqrt{t})$. This improves upon a result of Ezra and Lovett for this range of parameters. 2. In $mathcal{H}_2$, when $p= frac{1}{2}$, and $n = Omega(m log m)$, we show that the discrepancy is almost surely at most $1$. This answers an open problem of Hoberg and Rothvoss.
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