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2. Turán functions for books

Let the book $B_{k,m}$ consist of $m$ edges sharing $k-1$ common points plus one more edge that contains the remaining $m$ points and is disjoint otherwise. Let us exclude the case $m\le 1$ when $\pi(B_{k,m})=0$. The hypergraph problems for books turned out (relatively) more tractable. We know $\operatorname{ex}(n,B_{k,m})$ exactly for all large $n$, when $2\le m\le k\le 4$, see [<a href="#" class="cite">bollobas:74,frankl+furedi:83,furedi+pikhurko+simonovits:03:ejc,keevash+mubayi:04,furedi+pikhurko+simonovits:05,furedi+pikhurko+simonovits:06,furedi+mubayi+pikhurko:08,pikhurko:08:c</a>].
    1.     Frankl and Füredi [frankl+furedi:89] determined $\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by blowing up a small design. The following question is still open (see Frankl and Füredi \cite[Conjecture 1.5]{frankl+furedi:89}):

      Problem 2.1.

      Determine $\operatorname{ex}(n,B_{5,2})$ and $\operatorname{ex}(n,B_{6,2})$ exactly for all large $n$.
        1. Remark. One difficulty for this problem is that it is not clear how to prove the stability property, that is, that all almost extremal graphs have similar structure.
            • Conjecture 2.2.

              [[furedi+mubayi+pikhurko:08,bohman+frieze+mubayi+pikhurko:10]] \[ \pi(B_{5,5})=\frac{40}{81}, \] and \[ \pi(B_{6,6})=\frac12. \]
                1. Remark. The lower bounds come from a “bipartite” construction. It was proved in [bohman+frieze+mubayi+pikhurko:10] that $\pi(B_{5,5})\le 0.534...$ and that the bipartite construction is not optimal for $\pi(B_{k,k})$ when $k\ge 7$.
                    • Remark. The Turán density is unknown for $B_{5,3}$ and $B_{5,4}$ which is an interesting (and perhaps tractable) open problem.

                          Cite this as: AimPL: Hypergraph Turán problem, available at http://aimpl.org/hypergraphturan.