Saturation in Random Hypergraphs

TL;DR

This work extends saturation and weak saturation results from graphs to r-uniform hypergraphs in the binomial random host G^r(n,p). For fixed 2 ≤ r < s and constant p, it proves that whp wsat(G^r(n,p),K^r_s) equals wsat(K^r_n,K^r_s), while sat(G^r(n,p),K^r_s) is (1+o(1)) times the hypergraph-appropriate factor binom{n}{r-1} with a logarithmic term, reflecting the extra blow-up in the strong case. The authors develop a hypergraph core-activation framework to construct a weakly saturated subhypergraph H of G with size close to the complete-hypergraph benchmark and prove that all remaining edges can be activated via a carefully orchestrated sequence using auxiliary cores. For strong saturation, they adapt the Korándi–Sudakov approach with hypergraph-specific lemmas to obtain upper bounds through a multi-layer partition and activation strategy. Overall, the results establish the analogue of the graph saturation phenomena in random hypergraphs and illuminate the separate behaviors of weak versus strong saturation in this setting.

Abstract

Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n]:=\{1,2,...,n\}$ and whose edge set is $\binom{[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$ independently with probability $p$. An $r$-uniform hypergraph $H\subseteq G$ is $F$-saturated if $H$ does not contain any copy of $F$, but any missing edge of $H$ in $G$ creates a copy of $F$. Furthermore, we say that $H$ is weakly $F$-saturated in $G$ if $H$ does not contain any copy of $F$, but the missing edges of $H$ in $G$ can be added back one-by-one, in some order, such that every edge creates a new copy of $F$. The smallest number of edges in an $F$-saturated hypergraph in $G$ is denoted by $sat(G,F)$, and in a weakly $F$-saturated hypergraph in $G$ by $wsat(G,F)$. In 2017, Korándi and Sudakov initiated the study of saturation in random graphs, showing that for constant $p$, with high probability $sat(G(n,p),K_s)=(1+o(1))n\log_{\frac{1}{1-p}}n$, and $wsat(G(n,p),K_s)=wsat(K_n,K_s)$. Generalising their results, in this paper, we solve the suturation problem for random hypergraphs for every $2\le r < s$ and constant $p$.

Figures (3)

• Figure 1: Illustration of a chain of cores, $C_0$, $C_{v_2}$, and $C_{\{v_1,v_2\}}$, together with edges that were added to $H$. The edge $\{v_1, v_2, c_1\}$ is added to $H$ by \ref{['step v1,v2 and cs']}. The edge $\{v_2, c_1, c_2\}$ is added to $H$ by \ref{['step v1,v2 and cs']}, noting that $C_{\{v_2, c_1\}}=C_{v_2}$ by \ref{['observation: 3']}. The edge $\{c_1, c_2, c_3\}$ is added to $H$ by \ref{['step v1,v2 and cs']} since $C_{\{c_1, c_2\}}=C_0$ by \ref{['observation: 5']}. Finally, the edge $\{v_2, c_2, c_3\}$ is added to $H$ by \ref{['step v, c0 and cv']}.
• Figure 2: The edge $e$ closes a copy with $C_0$ and can thus be activated. The three types of edges that are in $H$ appear in shaded colours. The edges induced by $C_0$ appear in blue, and were added to $H$ at Step \ref{['step c0']}. The edges that contain one vertex from $e$ and two vertices from $C_0$ appear in red, and were added to $H$ at Step \ref{['step v and cv']}. The edges that contain two vertices from $e$ and one vertex from $C_0$ appear in green, and were added to $H$ at Step \ref{['step v1,v2 and cs']}.
• Figure 3: Illustration of a chain of activation, with the complexity of the construction evident already when $r=3$. Towards activating an edge $\{v_1, v_2, v_3\}$ with $\Tilde{C}$, we need to activate edges of the form $\{v_1, v_2, c\}$ where $c\in \tilde{C}$. To that end, we first activate all edges that form a clique with $C_0$, and in particular, all edges induced by $C_{v_1}\cup C_{\{v_1,v_2\}}\cup \tilde{C}$. Then, as the left side illustrates, we can activate the edges of the form $\{v_1, c_1, c_2\}$ as they form a clique with $C_{v_1}$. Indeed, the edge $\{c_1, c_2, c_3\}$ forms a clique with $C_0$, and the edge $\{v_1, c_2, c_3\}$ is in $H$ since $C_{\{v_1, c_2\}}=C_{v_1}$ (and was added at Step \ref{['step v1,v2 and cs']}). We can then, as the right side illustrates, turn our attention to edges of the form $\{v_1, v_2, c\}$, which will close a clique with $C_{\{v_1, v_2\}}$. Here, the edge $\{c, c_1, c_2\}$ closes a clique with $C_0$ and thus has already been activated, and the edge $\{v_1, v_2, c_2\}$ is in $H$ by Step \ref{['step v1,v2 and cs']}.

Theorems & Definitions (39)

• Theorem 1.1: A85F82K84K85L77
• Theorem 1.2: KS17
• Theorem 1
• Claim 3.1
• proof
• Lemma 3.2
• proof
• Claim 3.3
• proof
• Claim 3.4