Extremal Problems for Finite Sets
by Peter Frankl and Norihide Tokushige

Errata

1. page 206, line 5, in the RHS of the polynomial \( f_y \) the second sum should be taken over \( I, K \) instead of \( J, K \). On the same page there are some \( x, y, z\) which should be written in the math bold face.
2. page 7, line 13, "\( S_x (S_y (\mathcal F)) = S_y (S_x (\mathcal F)) \)" is wrong. In fact, if \( \mathcal F = \{ \{x,y,z\}, \{z\} \} \) then \( S_x (S_y (\mathcal F)) = \{ \{x,z\}, \{z\} \} \) while \( S_y (S_x (\mathcal F)) = \{ \{y,z\}, \{z\} \} \). The point here is that if \( x \in F \in S_x(\mathcal F) \) then \( F \setminus \{x\} \in S_x(\mathcal F) \), and moreover this property is unchanged even if we replace \( S_x(\mathcal F) \) with \( S_y(S_x(\mathcal F)) \). Noted by Bjarne Schülke, 2019.07.26.
3. page 8, proof of Proposition 2.6, replace \( \mathcal F \) and \( S_x(\mathcal F) \) with \( \mathcal F|_T \) and \( S_x(\mathcal F)|_T \), respectively. Noted by Bjarne Schülke, 2019.07.29.
4. page 8, line -2, delete "\( \in \)"
5. page 13, in the proof of Theorem 3.1, "\( X \)" should be "\( [n] \)" (two places).
6. page 15, line 3--5, "For any \(\mathcal F\subset 2^X\) ... is not true" should be "If \(\mathcal F\subset 2^X \) satisfies that \( |\mathcal F|_Y|=t \) for all \( Y \in \binom Xy \), then \( (|X|,|\mathcal F|) \rightarrow (y,t+1) \) is not true." Noted by Bjarne Schülke, 2019.07.29.
7. page 21, line 9, at the end of the sentence, $\mathcal G(n)$ should be $\mathcal G(\bar n)$.
8. page 21, proof of Theorem 4.5: The induction is written in a confusing way. The actual scheme should be as follows. Let $S(n,s)$ be the statement that "if $|G\cup G'|\leq n-s$ holds for all $G,G'\in\mathcal G$ then (i) or (ii) holds." We will show $S(n,s)$ for $\{(n,s): n\geq 1,\, n\geq s\geq 1\}$. Note that for all $n\geq 1$ we have $S(n,n)$ and $S(n,1)$. In fact the latter is equivalent to Theorem 1.3 (i). Also $S(n,n-1)$ is true for all $n\geq 2$ (this is the $n=s+1$ case). Thus $S(3,s)$ is true for $1\leq s\leq 3$ at this point. Now choose $n\geq 4$ and then $1\leq s\leq n-1$ arbitrarily, and we show $S(n,s)$ by induction on $n$. Our assumption is that $S(n',s')$ is true for all $1\leq n'< n$ and $n' \geq s'\geq 1$. So we can apply $S(n-1,s-1)$ to $\mathcal G(\bar n)$ and $S(n-1,s+1)$ to $\mathcal G(n)$. Noted by Bjarne Schülke, 2019.07.29.
9. page 10, the proof of Theorem 2.14: The first two lines of the proof should be replaced with the following. "We prove the statement by induction on $k$ and $x$. One can verify the statement for the case $k=1$ (and $x\geq k$). Let $k\geq 2$, and assume that the statement holds for all $(k',x')$ with $k'\leq x'$ and $k' < k$. We show the statement for $(k,x)$ with $x\geq k$. The case $x=k$ is trivial, so we suppose $x\geq k+1$, that is," Noted by Bjarne Schülke, 2019.08.04.
10. page 24, line -5: $\mathcal F=1,2,1,3,2,3$ should be $\mathcal F=\{\{1,2\},\{1,3\},\{2,3\}\}$.
11. page 26, line -5: at the end of the displayed equation should be "$\ldots\leq \displaystyle \sum_{S\in\mathcal S}\frac 1 {\binom n{|S|}}\leq 1$" instead of "$\ldots\leq\displaystyle\sum_{S\in\mathcal S} \frac 1 {\binom n{\lfloor\frac n{|S|}\rfloor}}\leq 1$."
12. page 28, line -13: "Let $t\geq 2$" should be "Let $r\geq 2$."
13. page 28, line -3: "arcs $F_1^{(j)},\ldots,F_r^{(j)},$ where ..." should be "arcs $F_1^{(j)},\ldots,F_r^{(j)}$ of size $k_1,\ldots,k_r$, respectively, where ..."
14. page 29, between line 13 and 14: add "If $r=2$ then we may assume that $k_1=1, k_2=n$, but in this case $\mathcal F_1$ and $\mathcal F_2$ cannot be 2-cross union. So we may assume that $r\geq 3$." Noted by Bjarne Schülke, 2019.08.04.
15. page 29, line 14: "assume that $|F_s| < n$" should be "assume that $|\mathcal F_s| < n$."
16. page 31, line -3: "$A=\{2,7.10\}$" should be "$A=\{2,7,10\}$."
17. page 32, line -2: "$\binom{[n]}k$" should be "$\binom{[a_k]}k$."
18. page 33, line -8: the first term in the RHS of the displayed equation should be "$\binom{a_{k-1}}{k-2}$" instead of "$\binom{a_{k-1}}{k-1}$."
19. page 35, line 6: "$|\mathcal F(1)| \leq \ldots$" should be "$|\mathcal F(1)| \geq \ldots$"
20. page 35, line 12: "then (b) and the induction hypothesis ..." should be "then the induction hypothesis and (a) ..."
21. page 41, line 5: "Let $l,s,m$, ... integers." should be "Let $l,s,m$, ... integers with $n=ms$."
22. page 41, at the end of the last sentence: "$1\leq i\leq 2(m-l)+1$" should be "$1\leq i\leq 2(m-l)-1$."
23. page 125, line 10: the last term is $(N-v+1)$ instead of $(N-v-1)$. (2021.02.22)
24. page 55, (10.2): the RHS is $(k-t+1)(2+\frac{t-1}{r})$ instead of $(k-t+1)(2+\frac{r-1}r)$.
25. page 57, in the middle, displayed inequlaity: the RHS is $(k'-t'+1)(2+\frac{t'-1}{r'})$ instead of $(k'-t'+1)(2+\frac{r-1}r')$. (2022.09.28)
26. page 198, line $-9$: "with new $n\times n$ matrices" should be "with new $2^n\times 2^n$ matrices." (2023.07.09)

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