A skew Bollobás system <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {P}=\{(A_i,B_i):1\le i\le m\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>B</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is a collection of pairs of disjoint subsets of [n] such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A_i\cap B_j\ne \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>∩</mo> <msub> <mi>B</mi> <mi>j</mi> </msub> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$1\le i<j\le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>. Denote by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_1(a, b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S_2(a, b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the maximum size of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\bigcup _{i=1}^m A_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>⋃</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <msub> <mi>A</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\bigcup _{i=1}^m B_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>⋃</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <msub> <mi>B</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, respectively, over all possible skew Bollobás systems <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {P}=\{(A_i,B_i):1\le i\le m\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>B</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$|A_i| \le a$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>a</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$|B_i| \le b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>b</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$i \in [m]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. It is shown that for any non-negative integers a and b, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$S_1(a,b)=\left( {\begin{array}{c}a+b+1\\ a\end{array}}\right) -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>a</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$S_2(a,b)=\left( {\begin{array}{c}a+b+1\\ a+1\end{array}}\right) -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

The maximum size of the partial ground set of skew Bollobás systems

期刊论文
发表日期：

2026年2月1日

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Yu Fang,
Tao Feng,
Xiaomiao Wang

摘要

A skew Bollobás system $\mathcal {P}=\{(A_i,B_i):1\le i\le m\}$ $P = {(A_{i}, B_{i}) : 1 \leq i \leq m}$ is a collection of pairs of disjoint subsets of [n] such that $A_i\cap B_j\ne \emptyset $ $A_{i} \cap B_{j} \neq \emptyset$ for any $1\le i<j\le m$ $1 \leq i < j \leq m$ . Denote by $S_1(a, b)$ $S_{1} (a, b)$ or $S_2(a, b)$ $S_{2} (a, b)$ the maximum size of $\bigcup _{i=1}^m A_i$ $⋃_{i = 1}^{m} A_{i}$ or $\bigcup _{i=1}^m B_i$ $⋃_{i = 1}^{m} B_{i}$ , respectively, over all possible skew Bollobás systems $\mathcal {P}=\{(A_i,B_i):1\le i\le m\}$ $P = {(A_{i}, B_{i}) : 1 \leq i \leq m}$ satisfying $|A_i| \le a$ $| A_{i} | \leq a$ and $|B_i| \le b$ $| B_{i} | \leq b$ for all $i \in [m]$ $i \in [m]$ . It is shown that for any non-negative integers a and b, $S_1(a,b)=\left( {\begin{array}{c}a+b+1\\ a\end{array}}\right) -1$ $S_{1} (a, b) = (\begin{matrix} a + b + 1 \\ a \end{matrix}) - 1$ and $S_2(a,b)=\left( {\begin{array}{c}a+b+1\\ a+1\end{array}}\right) -1$ $S_{2} (a, b) = (\begin{matrix} a + b + 1 \\ a + 1 \end{matrix}) - 1$ .